batbot: a biologically inspired flapping and morphing bat robot
TRANSCRIPT
BaTboT: a biologically inspired flapping and
morphing bat robot actuated by SMA-based
artificial muscles.
Julian David Colorado M.
Department of Electronics, Informatics and Industrial Engineering
Universidad Politecnica de Madrid, Spain
A thesis submitted for the degree of
Doctor of Philosophy in Robotics
2012
Title:
BaTboT: a biologically inspired flapping and morphing bat robot actuated by
SMA-based artificial muscles.
Author:
Julian David Colorado M, M.Sc
Director:
Prof. Antonio Barrientos Cruz, Ph.D
Prof. Claudio Rossi, Ph.D
Robotics and Cybernetics Group
Tribunal nombrado por el Mgfco. y Excmo. Sr. Rector de la Universidad Politectica de
Madrid, el dıa ........ de ........ de 2012.
Tribunal
Presidente:
Vocal:
Vocal:
Vocal:
Secretario:
Suplente:
Suplente:
Realizado el acto de lectura y defensa de la Tesis el dıa ........ de .......de 2012.
Calificacion de la Tesis: .......
El presidente: Los Vocales:
El Secretario:
ii
Abstract
Bats are animals that posses high maneuvering capabilities. Their wings contain
dozens of articulations that allow the animal to perform aggressive maneuvers by
means of controlling the wing shape during flight (morphing-wings). There is no
other flying creature in nature with this level of wing dexterity and there is biological
evidence that the inertial forces produced by the wings have a key role in the attitude
movements of the animal.
This can inspire the design of highly articulated morphing-wing micro
air vehicles (not necessarily bat-like) with a significant wing-to-body
mass ratio.
This thesis presents the development of a novel bat-like micro air vehicle (BaTboT )
inspired by the morphing-wing mechanism of bats. BaTboT’s morphology is alike
in proportion compared to its biological counterpart Cynopterus brachyotis, which
provides the biological foundations for developing accurate mathematical models
and methods that allow for mimicking bat flight.
In nature bats can achieve an amazing level of maneuverability by combining flap-
ping and morphing wingstrokes. Attempting to reproduce the biological wing ac-
tuation system that provides that kind of motion using an artificial counterpart
requires the analysis of alternative actuation technologies more likely muscle fiber
arrays instead of standard servomotor actuators. Thus, NiTinol Shape Memory Al-
loys (SMAs) acting as artificial biceps and triceps muscles are used for mimicking
the morphing wing mechanism of the bat flight apparatus. This antagonistic con-
figuration of SMA-muscles response to an electrical heating power signal to operate.
This heating power is regulated by a proper controller that allows for accurate and
fast SMA actuation. Morphing-wings will enable to change wings geometry with
the unique purpose of enhancing aerodynamics performance. During the downstroke
phase of the wingbeat motion both wings are fully extended aimed at increasing the
area surface to properly generate lift forces. Contrary during the upstroke phase
of the wingbeat motion both wings are retracted to minimize the area and thus
reducing drag forces.
Morphing-wings do not only improve on aerodynamics but also on the inertial forces
that are key to maneuver. Thus, a modeling framework is introduced for analyzing
how BaTboT should maneuver by means of changing wing morphology. This allows
the definition of requirements for achieving forward and turning flight according
to the kinematics of the wing modulation. Motivated by the biological fact about
the influence of wing inertia on the production of body accelerations, an attitude
controller is proposed. The attitude control law incorporates wing inertia informa-
tion to produce desired roll (φ) and pitch (θ) acceleration commands. This novel
flight control approach is aimed at incrementing net body forces (Fnet) that generate
propulsion.
Mimicking the way how bats take advantage of inertial and
aerodynamical forces produced by the wings in order to both increase
lift and maneuver is a promising way to design more efficient
flapping/morphing wings MAVs. The novel wing modulation strategy
and attitude control methodology proposed in this thesis provide a
totally new way of controlling flying robots, that eliminates the need of
appendices such as flaps and rudders, and would allow performing
more efficient maneuvers, especially useful in confined spaces.
As a whole, the BaTboT project consists of five major stages of development:
• Study and analysis of biological bat flight data reported in specialized
literature aimed at defining design and control criteria.
• Formulation of mathematical models for: i) wing kinematics, ii) dynamics,
iii) aerodynamics, and iv) SMA muscle-like actuation. It is aimed at modeling
the effects of modulating wing inertia into the production of net body forces
for maneuvering.
• Bio-inspired design and fabrication of: i) skeletal structure of wings and
body, ii) SMA muscle-like mechanisms, iii) the wing-membrane, and iv) elec-
tronics onboard. It is aimed at developing the bat-like platform (BaTboT)
that allows for testing the methods proposed.
• The flight controller: i) control of SMA-muscles (morphing-wing modula-
tion) and ii) flight control (attitude regulation). It is aimed at formulating
the proper control methods that allow for the proper modulation of BaTboT’s
wings.
• Experiments: it is aimed at quantifying the effects of properly wing modu-
lation into aerodynamics and inertial production for maneuvering. It is also
aimed at demonstrating and validating the hypothesis of improving flight effi-
ciency thanks to the novel control methods presented in this thesis.
This thesis introduces the challenges and methods to address these stages. Wind-
tunnel experiments will be oriented to discuss and demonstrate how the wings can
considerably affect the dynamics/aerodynamics of flight and how to take advan-
tage of wing inertia modulation that the morphing-wings enable to properly change
wings’ geometry during flapping.
Resumen:
Los murcielagos son mamıferos con una alta capacidad de maniobra. Sus alas estan
conformadas por docenas de articulaciones que permiten al animal maniobrar gracias
al cambio geometrico de las alas durante el vuelo. Esta caracterıstica es conocida
como (alas morficas). En la naturaleza, no existe ningun especimen volador con se-
mejante grado de dexteridad de vuelo, y se ha demostrado, que las fuerzas inerciales
producidas por el batir de las alas juega un papel fundamental en los movimientos
que orientan al animal en vuelo.
Estas caracterısticas pueden inspirar el diseno de un micro vehıculo
aereo compuesto por alas morficas con redundantes grados de libertad,
y cuya proporcion entre la masa de sus alas y el cuerpo del robot sea
significativa.
Esta tesis doctoral presenta el desarrollo de un novedoso robot aereo inspirado en
el mecanismo de ala morfica de los murcielagos. El robot, llamado BaTboT, ha sido
disenado con parametros morfologicos muy similares a los descritos por su sımil
biologico Cynopterus brachyotis. El estudio biologico de este especimen ha permi-
tido la definicion de criterios de diseno y modelos matematicos que representan el
comportamiento del robot, con el objetivo de imitar lo mejor posible la biomecanica
de vuelo de los murcielagos.
La biomecanica de vuelo esta definida por dos tipos de movimiento de las alas: ale-
teo y cambio de forma. Intentar imitar como los murcielagos cambian la forma de
sus alas con un prototipo artificial, requiere el analisis de metodos alternativos de
actuacion que se asemejen a la biomecanica de los musculos que actuan las alas, y
evitar el uso de sistemas convencionales de actuacion como servomotores o motores
DC. En este sentido, las aleaciones con memoria de forma, o por sus siglas en ingles
(SMA), las cuales son fibras de NiTinol que se contraen y expanden ante estımulos
termicos, han sido usados en este proyecto como musculos artificiales que actuan
como bıceps y trıceps de las alas, proporcionando la funcionalidad de ala morfica
previamente descrita. De esta manera, los musculos de SMA son mecanicamente
posicionados en una configuracion antagonista que permite la rotacion de las artic-
ulaciones del robot. Los actuadores son accionados mediante una senal de potencia
la cual es regulada por un sistema de control encargado que los musculos de SMA
respondan con la precision y velocidad deseada. Este sistema de control morfico de
las alas permitira al robot cambiar la forma de las mismas con el unico proposito de
mejorar el desempeno aerodinamico. Durante la fase de bajada del aleteo, las alas
deben estar extendidas para incrementar la produccion de fuerzas de sustentacion.
Al contrario, durante el ciclo de subida del aleteo, las alas deben contraerse para
minimizar el area y reducir las fuerzas de friccion aerodinamica.
El control de alas morficas no solo mejora el desempeno aerodinamico, tambien
impacta la generacion de fuerzas inerciales las cuales son esenciales para maniobrar
durante el vuelo. Con el objetivo de analizar como el cambio de geometrıa de las alas
influye en la definicion de maniobras y su efecto en la produccion de fuerzas netas,
simulaciones y experimentos han sido llevados a cabo para medir como distintos
patrones de modulacion de las alas influyen en la produccion de aceleraciones lineales
y angulares. Gracias a estas mediciones, se propone un control de vuelo, o control
de actitud, el cual incorpora informacion inercial de las alas para la definicion de
referencias de aceleracion angular. El objetivo de esta novedosa estrategia de control
radica en el incremento de fuerzas netas para la adecuada generacion de movimiento
(Fnet).
Imitar como los murcielagos ajustan sus alas con el proposito de
incrementar las fuerzas de sustentacion y mejorar la maniobra en
vuelo es definitivamente un topico de mucho interes para el diseno de
robots aeros mas eficientes. La propuesta de control de vuelo definida
en este trabajo de investigacion podrıa dar paso a una nueva forma de
control de vuelo de robots aereos que no necesitan del uso de partes
mecanicas tales como alerones, etc. Este control tambien permitirıa el
desarrollo de vehıculos con mayor capacidad de maniobra.
El desarrollo de esta investigacion se centra en cinco etapas:
• Estudiar y analizar el vuelo de los murcielagos con el proposito de definir
criterios de diseno y control.
• Formular modelos matematicos que describan la: i) cinematica de las alas,
ii) dinamica, iii) aerodinamica, y iv) actuacion usando SMA. Estos modelos
permiten estimar la influencia de modular las alas en la produccion de fuerzas
netas.
• Diseno y fabricacion de BaTboT: i) estructura de las alas y el cuerpo, ii)
mecanismo de actuacion morfico basado en SMA, iii) membrana de las alas, y
iv) electronica abordo.
• Contro de vuelo compuesto por: i) control de la SMA (modulacion de las
alas) y ii) regulacion de maniobra (actitud).
• Experimentos: estan enfocados en poder cuantificar cuales son los efectos
que ejercen distintos perfiles de modulacion del ala en el comportamiento
aerodinamico e inercial. El objetivo es demostrar y validar la hipotesis planteada
al inicio de esta investigacion: mejorar eficiencia de vuelo gracias al novedoso
control de orientacion (actitud) propuesto en este trabajo.
A lo largo del desarrollo de cada una de las cinco etapas, se iran presentando los retos,
problematicas y soluciones a abordar. Los experimentos son realizados utilizando
un tunel de viento con la instrumentacion necesaria para llevar a cabo las mediciones
de desempeno respectivas. En los resultados se discutira y demostrara que la inercia
producida por las alas juega un papel considerable en el comportamiento dinamico
y aerodinamico del sistema y como poder tomar ventaja de dicha caracterıstica para
regular patrones de modulacion de las alas que conduzcan a mejorar la eficiencia
del robot en futuros vuelos.
To God, my beloved father, his blessing is the reason for who I am nowadays. To
my parents, German and Patricia, who have always loved and supported me
during my whole life, their unconditional love, patience, wisdom and advising have
made me go beyond my aims. To my beloved brother Juan Felipe and Marita
because of their unconditional love and tenderness throughout my life. To my
advisors and friends, Antonio Barrientos and Claudio Rossi, who have offered me
constructive advice and helped me focus on feasible subjects to deal within this
endeavor of making this research successful. Finally, to all friends and fellows,
thank you very much, and The Lord bless you....
Acknowledgements
Words are not enough to express my gratitude and to acknowledge to whom anyhow
contribute during the development of this endeavor research. The author would like
to thank to professors Kenny Breuer and Sharon Swartz for providing the support
and useful knowledge about the robot design, bat flight kinematics and aerody-
namics. To the Breuer Lab team for providing the wind-tunnel facility of Brown
University, USA and their support with the experiments. To the Robotics and
Cybernetics Group team for their warm collaborative environment and helpful dis-
cussions in general robotic sciences. Last but not least, to the Community of Madrid
and Universidad Politecnica de Madrid for their funding during the development of
this research.
Contents
List of Figures viii
List of Tables xxi
1 Introduction 1
1.1 The problem and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Original Contributions of this Work . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Literature Review 13
2.1 General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Nature flyers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Biomechanics: insects, birds, and bats . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Bat biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Bat flight research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Shape Memory Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Basic foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 Advantages and drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Morphing-wing MAVs with smart actuation . . . . . . . . . . . . . . . . . . . . . 32
2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 From bats to BaTboT: Mimicking biology 39
3.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Review on biological flight performance data . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Measurements of wing morphological parameters . . . . . . . . . . . . . . 40
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3.2.2 Measurements of kinematics parameters . . . . . . . . . . . . . . . . . . . 41
3.2.3 Measurements of Aerodynamics parameters . . . . . . . . . . . . . . . . . 42
3.3 Choice of species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Wing morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Biological-based framework for modeling and design . . . . . . . . . . . . 46
3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 BaTboT modeling 48
4.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Kinematics model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Wing and body kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.3 Wing trajectories and manuevers . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Inertial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 Spatial notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.2 Equations of motion (EoM) . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.3 Rolling and pitching torques . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Aerodynamics model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.1 Lift and drag forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.2 Net forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 SMA muscle-like wing actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.1 SMA actuation configurations . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5.2 SMA phenomenological model . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Simulation and experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6.1 Open-loop simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.6.2 Wing torques for actuation . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.6.3 Body torques for maneuvering: experiments for inertial model validation . 82
4.6.4 SMA-muscle limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 BaTboT design and Fabrication 90
5.1 The general method for BaTboT’s design . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Prototype characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 Design process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
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CONTENTS
5.2.2 Components and weight distribution . . . . . . . . . . . . . . . . . . . . . 92
5.3 BaTboT mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.1 Step 2: Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1.1 Bio-inspired parameters . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1.2 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 Step 3: Fixed-wing design . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3.2.1 Flapping-wing mechanism . . . . . . . . . . . . . . . . . . . . . . 101
5.3.2.2 Membrane issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3.3 Step 4: Articulated-wing design . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.4 Step 5: Morphing-wing mechanism . . . . . . . . . . . . . . . . . . . . . . 105
5.3.5 Step 6: The wing-membrane . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 BaTboT electronics and sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4.1 Arduino controller-board . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4.2 The Inertial Measurement Unit (IMU) . . . . . . . . . . . . . . . . . . . . 113
5.4.3 SMA power drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.5 BaTboT consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.6 BaTboT costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 BaTboT Control 118
6.1 Control goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Flight Control Architecture (FCA) . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 SMA actuation: experimental characterization . . . . . . . . . . . . . . . . . . . 122
6.3.1 Frequency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3.2 Experimental validation of SMA actuation model . . . . . . . . . . . . . . 126
6.3.3 Data summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 SMA Resistance-to-Motion relationship (RM) . . . . . . . . . . . . . . . . . . . . 131
6.5 Morphing-wing control (inner loop) . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.5.1 Sliding-mode control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.5.2 PID control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.5.3 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5.4 Morphing-wing control algorithm . . . . . . . . . . . . . . . . . . . . . . . 136
6.6 Attitude control (outer loop) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.6.1 Backstepping+DAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
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CONTENTS
6.6.2 Attitude control algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.7 Simulation and experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.7.1 Variables and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.7.2 Morphing-wing control response . . . . . . . . . . . . . . . . . . . . . . . 145
6.7.3 Attitude control response . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.8 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7 General experimental results 154
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.1.1 Methods and goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.1.2 The wind-tunnel setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2 Control performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2.1 Morphing-wing control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2.1.1 SMA accuracy and speed . . . . . . . . . . . . . . . . . . . . . . 158
7.2.1.2 SMA fatigue issues . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2.2 Attitude control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2.2.1 Forward flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2.2.2 Turning flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3 Aerodynamics experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.4 Efficient Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.5 Discussion of results: Towards efficient flight . . . . . . . . . . . . . . . . . . . . . 168
7.5.1 Morphing-wing modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.5.2 Wing inertia for efficient flight . . . . . . . . . . . . . . . . . . . . . . . . 168
8 Conclusions and Future Work 170
8.1 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.3 Thesis schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9 Publications 174
9.1 Journals, book chapters and conference proceedings . . . . . . . . . . . . . . . . 174
9.2 Press and media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
References 176
vi
CONTENTS
10 Annexes 180
10.1 Floating-base forward kinematics Matlab-code . . . . . . . . . . . . . . . . . . . . 180
10.2 Floating-base inverse dynamics Matlab-code . . . . . . . . . . . . . . . . . . . . . 181
10.3 Floating-base forward dynamics Matlab-code . . . . . . . . . . . . . . . . . . . . 181
10.4 SMA phenomenological model Matlab-code . . . . . . . . . . . . . . . . . . . . . 182
10.5 Control code programmed into the Arduino . . . . . . . . . . . . . . . . . . . . . 184
10.6 Flight Control Matlab environment . . . . . . . . . . . . . . . . . . . . . . . . . . 189
vii
List of Figures
1.1 BaTboT. The overall mass of the skeleton+electronics+battery is 125g. The wingspan:
53cm (wings fully extended). Each wing of the robot has six degrees of freedom (dof):
2-dof at shoulder, 1-dof at elbow, and 3-dof at wrist joint. The body frame {b} is a
6-dof floating body. Rotations about the body-frame {b}-xb, yb, zb axes are designated
roll, pitch and yaw following aerodynamic conventions. Frame {o} is the inertial frame. 2
1.2 a) bats are agile flyer for hunting preys in air, water or even ground, b) bats
can hover like hummingbirds, consuming less energy (10 − 20Hz of flapping
frequency), c) bat wings can camber, stretch, extend and fold like no other
flying animal in nature, d) VTOL flight capacity, Source: Breuer Lab, http:
//brown.edu/Research/Breuer-Lab/research/batflight.html . . . . . . . . 5
1.3 Structural steps to be followed during the thesis aimed at the development of
BaTboT. The pictures depicted herein, correspond to the final BaTboT proto-
type. The forthcoming chapters will introduce each step with all the details.
Source: The author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Structure of animal wings showing the main skeletal support. (Left) vertebrates,
(Right) insects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Comparison for several species of small-scale flyers: a) Wingspan, b) Aspect
ratio, c) Wing loading, d) Wingbeat frequency. Source from (1), (2), (3). . . . . . 16
2.3 a) Cartoon of aerodynamics forces acting on a typical wing design, b) Lift and
drag changes with the angle of attack for a typical wing design. c) Comparison
of the lift to drag ratio. d) Power Flight. Source from (1), (2), (3). . . . . . . . . 18
2.4 Bat anatomy. A) body anatomy, B) Pectoral skeleton, C) Dorsal view of re-
tracted right arm, D) Dorsal view of expanded right arm. Source (4). . . . . . . . 20
2.5 Bat muscle structure to power wingstroke motion. . . . . . . . . . . . . . . . . . 21
viii
LIST OF FIGURES
2.6 Bat wing membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 Tracking results of a bat using a 52 degree of freedom articulated model. Shown
on top are frames extracted from high speed video of a landing bat. Shown on
the bottom are the corresponding frames of the reconstructed three-dimensional
wing and body kinematics. –Caption extracted from (5)–. . . . . . . . . . . . . . 23
2.8 Reconstruction of the wake of C. brachyotis bat. Source: (6). . . . . . . . . . . . 24
2.9 Strain measurements of bat’s wing membrane. Source: (7). . . . . . . . . . . . . 25
2.10 Sensory wing hair before (A) and after (B and C) depilation. (A) Scanning
electron microscope image from a domed hair located on the ventral trailing
edge (location is marked by a gray circle in schematic to the Right) of Eptesicus
fuscus. Caption extracted from Source: (8). . . . . . . . . . . . . . . . . . . . . . 25
2.11 a) The hysteresis curve of SMA, b) stress-strain-temperature curve of SMA ex-
hibiting the one-way shape memory effect, c) stress-strain-temperature curve of
SMA exhibiting the two-way shape memory effect, d) stress-strain-temperature
curve of SMA exhibiting pseudoelasticity behavior. Source: (9). . . . . . . . . . . 27
2.12 a) The BATMAV robot, b) Detailed arm assembly using SMAs-bases muscles.
Source: (10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.13 a) Bat-wing platform in the wind-tunnel, b) Nylon membrane, c) Spandex mem-
brane, d) Silicone membrane. Source: (11). . . . . . . . . . . . . . . . . . . . . . 33
2.14 Harvard RoboBee, actuated by PZT. Source: (12). . . . . . . . . . . . . . . . . . 34
2.15 Inner structure and possible control modes of a multi-functional trailing edge.
Source: (13). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.16 Layout of an MFC actuation device for MAV morphing-wing camber. Source:
(14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.17 TOP (left to right) Smart-bird by FESTO, inspired by the Seagles (15), Combat
by University of Michigan, inspired by bat’s navigation system (16), DARPA
nano-air hummingbird (http://www.darpa.mil). BELOW (left to right) The
Gull Wing by (17), MFX-1 by NextGen Aeronautics, inspired by the batwing
internal structure (18), Prototype of Entomoter MAV (19). . . . . . . . . . . . . 36
ix
LIST OF FIGURES
3.1 Right lateral view of the bat with respect to the inertial coordinate system {o}.Green dots are the path of the wrist joint whereas red dots are the path of the
wingtip over a wingbeat cycle. The position and posture of the right wing are
shown at three time points in the wingbeat cycle. Source: (20). . . . . . . . . . 40
3.2 (A) Maximum wingspan, (B) minimum wingspan, (C) wing chord, (D) maximum
wing area, (E) wing loading, and (F) aspect ratio. Circles represent medians for
each species and the black arrow points to the specimen under analysis. Source:
(20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 (A) Flight speed, (B) Horizontal accelerations, (C) Vertical accelerations. Circles
represent medians for each species and the black arrow points to the specimen
under analysis. Source: (20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Wingbeat period (A) scaled lower than expected under isometry. Downstroke
duration (B), Downstroke ratio (C), stroke amplitude (D), stroke plane angle (E)
and Strouhal number (F) did not change significantly with body mass. Angle
of attack increased with body size (G) as a result of a change in α1 (H), but
not from a change in α2 (I). Wing camber (J) did not change with body size,
but coefficient of lift (K) did. Circles represent medians for each species and the
black arrow points to the specimen under analysis. Source: (20). . . . . . . . . . 43
3.5 Bioinspired MAV classification depending on wingspan and mass. Source: the
author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Wing physiology. 17 markers are placed on: anterior and posterior sternum (a
and b, respectively), shoulder (c), elbow (d), wrist (e), the metacarpophalangeal
and interphalangeal joints and tips of digits III (f, g, h), IV (i, j, k), and V (l,
m, n), the hip (o), knee (p), and foot (q). Source: (21). . . . . . . . . . . . . . 44
3.7 Detailed parameters that describe wing segment morphology. a) wing segment
subdivision, b) detailed configuration of the wrist joint and attached digits when
the wing is fully extended. It shows the angles between digits that maintain
proper wing membrane tension during downstroke. Source: the author. . . . . 45
4.1 Topology. The robot has an overall of 14-DoF (not counting the 6-DoF of the
floating body). Each wing has 6-DoF and each leg 1-DoF. Source: the author. 51
x
LIST OF FIGURES
4.2 Detailed description of wing kinematics frames based on Denavit-Hartenberg
(DH) convention, qi corresponds to the rotation angle from axis xi−1 to xi mea-
sured about zi. The subscript i indicates the frame of reference (i = 1..6). The
inset is a top view of the right wing showing planar angles. . . . . . . . . . . . . 52
4.3 a)-b) Example of Cartesian paths during turning and forward flight, c-d) exam-
ple of wing modulation scheme for wing contraction during upstroke and wing
extension during downstroke. Source: the author. . . . . . . . . . . . . . . . . 55
4.4 Stills of wing kinematics during forward flight: a) beginning of downstroke, b)
middle downstroke, c) end of downstroke/beginning upstroke, d) middle up-
stroke. Source: (20), (22), (23), (24). . . . . . . . . . . . . . . . . . . . . . . 56
4.5 (Simulation) Wing kinematics of forward flight (both wings move symmetrically
at wingbeat frequency of f = 2.5Hz): a) Cartesian trajectories of the wrist joint
and the wingtip frame. b) Joint angles. c)-d) Joint velocities and accelerations.
For velocities, q3 = q4 (red plot) and q5 = q6 (purple plot). For accelerations,
q3 = q4, q5 = q6. Source: the author. . . . . . . . . . . . . . . . . . . . . . . . 57
4.6 (Simulation) Wing kinematics of turning flight at wingbeat frequency of f =
2.5Hz): a) Cartesian trajectories for the wrist joint and the wingtip frame of
each wing (top view). b)-c)-d) Left wing more contracted: joint angles, velocities
and accelerations. e)-f)-g) Right wing more extended: joint angles, velocities and
accelerations. For velocities, q3 = q4 (red plot) and q5 = q6 (purple plot). For
accelerations, q3 = q4, q5 = q6. Source: the author. . . . . . . . . . . . . . . . 59
4.7 Comparison between wing joint trajectories of left and right wings (q3 and q4)
considering a morphing-wing factor of fmc = 0.1. Source: the author. . . . . . 60
4.8 Rigid multibody serial chain that composes each wing. Spatial forces of each body
contain both linear fi and angular τi force components stacked into a six-dimensional
vector Fi. These forces are propagated from the wingtip {i} = n to the base frame
{0}. Subscripts R,L denote for right and left wing respectively. The resultant spatial
force (FT ) acting on the base frame {0} is the sum of spatial forces generated by both
wings. The inset shows the velocity of a rigid body i expressed in terms of ωi and vi,
and the force acting on a rigid body i expressed in terms of fi and τi. . . . . . . . . . 61
xi
LIST OF FIGURES
4.9 Free-body diagram of a bat in accelerating flight, indicating the aerodynamic
and gravitational forces that accelerate the center of mass (COM). Lift is per-
pendicular to the direction of flight whereas drag and thrust are parallel to the
direction of flight. The net force produced can be decomposed into net force
components parallel and perpendicular to the direction of flight (see inset). The
parallel component corresponds to the net thrust. Thus, measurements of the ac-
celeration of the COM would directly reflect the net forces acting on it. Source:
(24). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.10 (Experimental) aerodynamics identification –wind-tunnel–: a) the bat-robot is
mounted on top of a 6-DoF force sensor from which both lift FL and drag FD
forces are experimentally calculated as a function of the airflow speed and angle
of attack (AoA), b) Lift and drag coefficients (CL, CD) calculated from measured
lift and drag forces. Source: the author. . . . . . . . . . . . . . . . . . . . . . 67
4.11 SMA actuation configurations: a) (top) SMA joint with bias spring concept,
(medium) mechanic implementation, (below) SMA-spring model representation.
b) (top) SMA antagonistic joint, (medium) mechanic implementation that mimic
how bicep and tricep muscles operate, (below) antagonistic pair of SMAs model
representation. In both configurations SMA wires extend along the humerus
bone of BaTboT wings, acting as artificial muscles that pull the elbow joint q3.
SMA pulling forces (Fsma) produce a joint torque (τ3) that rotates the elbow,
pulling in the ”fingers” to slim the wing profile on the upstroke. Source: the
author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.12 SimMechanics open-loop simulator for dynamics and SMA actuation. Source:
the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.13 Wing torques denoted as τi correspond to the effective forces that the each joint
i requires to rotate as defined by trajectory profiles qi (see upper inset). Torques
are with respect to the joint frames {i} assigned by modified DH convention.
The lower inset shows the model for the estimation of torques as a function of
the joint profiles. The inertial model in Algorithm 1 takes into account robot
parameters/constrains and aerodynamic loads. Source: the author. . . . . . . 77
xii
LIST OF FIGURES
4.14 (Simulation) a) required wing torques (τi) during forward flight. Both wings flap
symmetrically describing the wing profiles qi shown in Figure 4.5b (wingbeat
frequency f = 5Hz), b) Increments of wingbeat frequency cause the wing torques
to increase. The plot shows maximum peak values of wing torques Source: the
author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.15 (Simulation) a) required wing torques (τi) during turning flight: a) left wing,
b) right wing. The upper plots show the difference between the wing torque
profile that each wing requires to describe the trajectory profiles qi shown in
Figure 4.6b-e respectively (wingbeat frequency f = 5Hz). The lower plots show
how increments of wingbeat frequency cause the wing torques of each wing to
increase. Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.16 (Simulation) effects of wing modulation on the production of body torques and
angular accelerations at wingbeat frequency of f = 1.3Hz a) wing modulation
profile qi for the left wing (upper) and right wing (lower), b) (upper) body torques
(τθ, τφ, τψ) produced at center of mass, (medium) body angular accelerations
(θ, φ, ψ) produced at center of mass, and (lower) attitude response of the robot
(θ, φ, ψ). Values are with respect to the body frame {b}, Source: the author. . 80
4.17 (Simulation) effects of wing modulation on the production of body torques and
angular accelerations at wingbeat frequency of a) f = 2.5Hz, b) f = 5Hz, and c)
f = 10Hz. (upper) body torques,(medium) body angular accelerations, (lower)
attitude response. Source: the author. . . . . . . . . . . . . . . . . . . . . . . 81
4.18 Effects of the flapping motion of the wings on the position of the center of mass (CM)
and accelerations of the body: a) end of upstroke motion, b) end of downstroke motion.
Upper plots depicts the testbed for experimental measurements of six-dimensional in-
ertial forces (FT ) and lower plots depicts simulations for the computation of inertial
forces (FT ). Values for both measured and simulated FT are consigned in following
Figure 4.19. To conserve momentum, the body moves in opposition to the flapping
direction. During upstroke, the upward and backward acceleration caused by the flap-
ping motions of the wings produce an inertial force (red circled arrow) that moves the
body forward and downward with respect to the downstroke. This force produces a
forward-oriented component (FT,x), or inertial thrust (green solid arrow). Contrary,
during the downstroke negative inertial thrust is produced. Source: the author. . . . 83
xiii
LIST OF FIGURES
4.19 (Experimental) wing inertia contribution on forward and turning flight: a) simulation
model VS experimental results of pitching torques τθ (f=5Hz), b) simulation model
VS experimental results of rolling torques τφ (f=5Hz), c)-d) quantification of wing
inertia contribution into the generation of τθ and τφ at different wingbeat frequencies
f . Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.20 (Simulation) pitching torque response (τθ) for different positions of q2 (wings
rotated forward/backward the body) and for different wingbeat frequencies f .
The set M1 contains bio-inspired wing modulation shown in Figure 4.5b whereas
set M2 corresponds to step-input signals that rotate each joint of the wing due
to the range defined in Table 4.6 (cf. joint rotation range). Source: the author. 85
4.21 (Simulation) rolling torque response (τφ) for different positions of q2 and wing-
beat frequencies f . The set M1 contains bio-inspired wing modulation whereas
set M2 corresponds to step-input signals that rotate each joint of the wing.
Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.22 (Simulation) SMA phenomenological model response at different current profiles: a)
Joint rotation based on SMA strain. b) Temperatures on the SMA wire, c) Hysteresis-
loop for the nominal operation mode (Isma = 350mA), d) SMA strain VS stress. . . . . 87
5.1 Design process of BaTboT. Source: The author. . . . . . . . . . . . . . . . . . . . 91
5.2 a) weight distribution of main components, b) detailed view of main components.
Source: The author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 a) Each morphological aspect of BaTboT design has been carefully approached
to its biological counterpart, b) Cartoon of typical wingtip trajectory described
during a wingbeat cycle. Source: The author. . . . . . . . . . . . . . . . . . . . . 96
5.4 a) Pin-out and Circuit Diagrams for SMA migamotor actuator model NM706-
Super, b) actuator dimensions. Source: The author. . . . . . . . . . . . . . . . . 99
5.5 (left) half fixed-wing flapping testbed, (right) experimental quantification of
power requirements for flapping at maximum f = 10Hz. Source: The author. . . 101
5.6 Detailed flapping-wing mechanism. Source: The author. . . . . . . . . . . . . . . 102
5.7 (left) platform for wing membrane fabrication, (right) membrane issues: air bub-
bles are kept within the mixture. Source: The author. . . . . . . . . . . . . . . . 103
xiv
LIST OF FIGURES
5.8 Articulated-wing design: a-b) metal tendons are placed inside the bones for
connecting the elbow joint with the wrist. This enables each digit rotate as a
function of the elbow’s rotation, c) cartoon of a biological contracted wing and
its parts, d) ABS fabricated articulated wing. Source: The author. . . . . . . . . 104
5.9 a) A high camera captures the motion of each marker places along the specime’s
wings. Pictures were taking during contraction and extension of the wings, b)
detailed dimensions of wing morphology during middle downstroke. Angles and
proportions are extracted from the in-vivo experiments from (a), c) membrane-
free maximum rotation of elbow and wrist joints during wing contraction and
extension, d) maximum rotation of elbow and wrist joints during wing contrac-
tion and extension including the wing-membrane load. Source: The author. . . . 105
5.10 Antagonistic mechanism of SMA-based muscle actuators. Source: The author. . 106
5.11 Testing on the stretchable property of the fabricated silicone membrane: a) ex-
tended plagiopatagium skin, embedded tiny muscles control the membrane ten-
sion, b) the fabricated silicone membrane is attached to the wing skeleton using
Sil-Poxy RTV, c) the fabrication process described in section 5.3.2.2 results on a
light artificial skin with the required stretchable property. Also note that after
vacuum, air-bubbles are eliminated. Source: The author. . . . . . . . . . . . . . . 108
5.12 Technical overview of Dragon Skin silicone properties. Source: http://www.
smooth-on.com. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.13 Mid-downstroke wing camber and angle of attack are estimated as follows: (A)
A parasagittal (xg-zg) cross section of the wing was taken at the yg-value of the
wrist at the time of maximum wingspan. Six triangular sections of the wing
membrane crossed that plane and the intersections of triangle borders in the
plane (red circles) were used as estimates of membrane position. (B) The actual
curved shape of the membrane in the plane (solid black line) was estimated
using the first term of a sine series fitted to those seven points. The maximum
distance of the membrane line from the chord line (dashed grey line) was divided
by the length of the chord line to give wing camber. (C) Angle of attack (α)
was calculated as α1 + α2, where α1 is the angle of the wing chord line above
horizontal (blue dashed line), and α2 is the angle between horizontal and the
velocity vector of the wrist (red arrow) in the xg − zg plane. Source: (20). . . . . 109
5.14 Variation of camber during the wingbeat cycle. Source: (25). . . . . . . . . . . . 110
xv
LIST OF FIGURES
5.15 Implications of wing camber into lift and drag production (Vair = 5ms−1, fixed-
wing): a) excessive wing camber (0.48), b) proper wing camber (0.16). Source:
The author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.16 Micro-controller and Migamotor SMA muscle connection diagram. Source: The
author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.17 Comparison of attitude IMU readings before and after Kalman filtering. Source:
Arduino. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.18 Miga analog driver V5 pinout diagram. Source: The author. . . . . . . . . . . . . 115
5.19 Percentage of current consumption per component. Source: The author. . . . . . 116
6.1 Flight Control Architecture (FCA). Source: The author. . . . . . . . . . . . . . . 119
6.2 Experimental testbed for the characterization of SMA input power (uheating) to
output torque (τ3). Forces (F ) are measured using a force sensor with 0.318 gram-
force of resolution. Torque conversion is applied by considering the humerus bone
length (lr), as: τ3 = Flr[Nm]. This allows the identification of SMA actuation.
Source: The author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3 (Experimental VS model) Bode magnitude and phase plots for NiTi 150μm SMA
Migamotor actuators. The insets show several experimental measurements of
magnitude and phase. Magnitude is given by: 20log(A/b), where A is the least-
squares estimation of the force amplitude measured using Eq. 6.3, and b is the
AC power of the input signal uheating. Phase is given by the term ϕ in Eq. 6.3.
The transfer function that fits the experimental data is shown in Eq. 6.4. This
plot also compares the model in Eq. 6.4 against the experimental data. Source:
The author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.4 (Above) right wing extended and contracted taking into account the load pro-
duced by the silicon membrane. (Below) Experimental measurements of SMA
output torque τ3 generated by input heating power uheating. Values are classified
by nominal and overloaded operation mode of the SMA actuators. Nominal be-
havior is achieved by applying uheating =∼ 1.36W whereas overloaded behavior
requires an input power of uheating =∼ 2.57W . Inset plots show the average peak
of produced elbow torque at both SMA operation modes. Source: The author. . 127
xvi
LIST OF FIGURES
6.5 Experimental measurements of joint rotation speeds of elbow joint (θ3) that are
obtained by applying heating power values at nominal (uheating = 1.36W ), and
overloaded (uheating = 3.06W ) SMA operation. Source: The author. . . . . . . . 128
6.6 (Experimental) Input Power to Output Torque small-signal response of the SMA
actuators, being uheating = a+ bsin(2πft), f = 2Hz. Source: The author. . . . . 129
6.7 (Experimental) Resistance-Motion (RM) linear relationship between SMA electrical
resistance change (Rsma) and the angular motion generated at the elbow joint (q3).
Small variations in ambient temperature (To) modify the RM relationship. The inset
shows BaTboT in the wind-tunnel. Ambient temperature has been measured with a MS
1000-CS-WC temperature sensor supplied by ATS (http://www.qats.com/Products).
Changes in Rsma are constantly measured during the experiment. Source: the author. . 130
6.8 General scheme for the inner morphing-wing control. Source: the author. . . . . . . . 132
6.9 Detailed inner loop of sliding-mode morphing-wing control with torque reference. Source:
the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.10 Detailed inner loop of PID morphing-wing control with joint position reference. Source:
the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.11 Typical elbow joint reference profile q3,ref during a wingbeat cycle (f = 1.25Hz). It
details how Algorithm 3 works. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.12 a) In-vivo recordings of a bat landing on the ceiling, cf. (26), b) closed-loop
control simulation of attitude maneuvering using the backsteping+DAF strategy.
Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.13 Detailed outer and inner loops: complete description of the Flight Control Architecture
(FCA). Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.14 Experimental setup using the wind-tunnel of Brown University: Flight Control Archi-
tecture (FCA). Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.15 (Simulation) morphing-wing based on sliding-mode response: a) control tracking given
a sinusoidal joint trajectory (q3,ref ) at wingbeat frequency of f = 4Hz, b) phase plane
of the sliding surface upon sinusoidal input from plot (a), c) electrical current (Isma)
to drive each SMA actuator in the antagonistic configuration. Source: the author. . . . 146
xvii
LIST OF FIGURES
6.16 (Experimental) morphing-wing based on sliding-mode response: a) SMA output torque
τ3 given a force reference τ3,ref , b) control tracking given a bio-inspired joint trajec-
tory q3,ref at wingbeat frequency of f = 2.5Hz; it shows comparison of sliding-mode
response with and without the saturation function sat(S) within the control law, c)
electrical current (Isma) to drive each SMA actuator in the antagonistic configuration
Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.17 (Simulation) morphing-wing based on PID response: a) PID control tracking given
a sinusoidal profile q3,ref (above), square profile (medium), sawtooth profile (below),
at wingbeat frequency of f = 2.5Hz, b) electrical current (Isma) to drive each SMA
actuator in the antagonistic configuration. Source: the author. . . . . . . . . . . . . . 148
6.18 (Experimental) morphing-wing based on PID response: a) measured SMA output
torque τ3, b) PID control tracking given a bio-inspired joint trajectory q3,ref , c) elec-
trical current (Isma) to drive each SMA actuator in the antagonistic configuration.
Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.19 (Simulation) backstepping+DAF attitude stabilization. Roll (φref ) and pitch (θref )
references are set to zero while initial attitude position is set to 20o and −20o for roll
and pitch respectively. Control efforts are also shown. Source: the author. . . . . . . . 151
6.20 (Experimental) backstepping+DAF attitude stabilization. Roll (φref ) and pitch (θref )
references are set to zero. The wind-tunnel airspeed has been set to 2ms−1 and the
controller must keep both angles close to zero. Source: the author. . . . . . . . . . . . 151
6.21 (Simulation) backstepping+DAF attitude tracking. Roll (φref ) and pitch (θref ) sinu-
soidal references are tracked. Source: the author. . . . . . . . . . . . . . . . . . . . . 152
6.22 (Experimental) backstepping+DAF attitude tracking. Roll (φ) and pitch (θ) profiles
are tracked for wind-tunnel airspeeds of 0 and 5ms−1. Tracking errors are measured.
Source: the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.1 a) setup for morphing-wing testing. b) wind-tunnel setup for dynamics, aerody-
namics and control testing. Source: the author. . . . . . . . . . . . . . . . . . . . 155
xviii
LIST OF FIGURES
7.2 Stills of morphing-wings’ control within the wind-tunnel. The wingbeat cycle is com-
posed by two phases: downstroke and upstroke. a) Beginning of the downstroke. The
body of the specimen is lined up in a straight line, elbow joint is ∼ 58o, b) end of
downstroke, the membrane is cambered and the wings are still extended, elbow joint
is ∼ 5o, c) middle of downstroke, the wings are extended to increase lift, elbow joint is
∼ 20o, d) upstroke, the wings are folded to reduce drag, elbow joint is ∼ 45o. Figures
a-b illustrate the process to measure aerodynamics loads using the force sensor located
at the center of mass of the robot (below the body). Figures c-d illustrate the process to
measure inertial forces at the center of mass produced by both wings (no aerodynamics
loads caused by the membrane). e-f) show the beginning of downstroke and the end of
the upstroke without the membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.3 (Experimental) morphing-wings’ control response. a) Tracking of the elbow’s joint
trajectory at f = 2.5Hz, Vair = 0m/s i.e., no-wind. b) Close-up to a wingbeat cycle.
The two plots describe the control tracking regarding: i) Vair = 0m/s (same than plot-
a), and ii) Vair = 5m/s. c) Electrical current Isma delivered to the antagonistic SMA
actuators, and regulated by the anti-slack and anti-overload mechanisms. d) Position
tracking errors from plot-b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.4 (Experimental) Performance of the SMA actuator for longer periods of actuation: a)
Nominal operation at 1.3[Hz], b) Overloaded operation at 2.5[Hz], c) Output torque
peaks extracted from overloaded response in plot-b). . . . . . . . . . . . . . . . . . . 159
7.5 Forward flight control. Backstepping+DAF attitude tracking at: a)-b) roll and pitch
tracking with Vair = 5ms−1, c)-d) roll and pitch tracking with Vair = 2ms−1. . . . . . 161
7.6 Turning flight control. Backstepping+DAF attitude tracking at: a)-b) roll and pitch
tracking with Vair = 5ms−1, c)-d) roll and pitch tracking with Vair = 2ms−1. . . . . . 162
7.7 (Experimental) Aerodynamics measurements. a) Comparison between lift and drag
coefficients (CL, CD) with and without the motion of the morphing-wings (Vair = 5m/s,
wingbeat frequency of f = 2.5Hz). b) Lift-to-drag ratio (L/D) as a function of the
angle of attack (Vair = 5m/s, f = 2.5Hz ). c) Wind-tunnel airspeed measurements
(Vair). d) Lift (L) and drag (D) forces corresponding to plot-a (with morphing). . . . . 163
7.8 (Forward flight) benefits of the DAF to the proper modulation of wing-morphology
aimed at incrementing net forces (f = 2Hz, φref = 0o, θref = 10o): a) without the
DAF, b) with the DAF. Top: attitude tracking error and disturbance rejection; middle:
detailed wing modulation (elbow joint q3); bottom: net forces generated. . . . . . . . 165
xix
LIST OF FIGURES
7.9 Effects of different wing modulation profiles (no-DAF terms presented) on lift and drag
production (f = 2.5Hz, Vair = 5ms−1): a) Cartesian trajectory of the wingtip gen-
erated during a wingbeat cycle measured with respect to the base frame {0}, b) wing
modulation profile of elbow joint q3 for different backstepping parameter values of λ2
and λ4 (no-DAF-1: λ2 = λ4 = 0.1, no-DAF-2: λ2 = λ4 = 0.05), c) wing modulation
profile of wrist joint q4 corresponding to the same backstepping parameter values con-
figuration from plot-b, d) (left) net forces and (right) lift and drag coefficients generated
with the backstepping parameter configuration no-DAF-1, e) left) net forces and (right)
lift and drag coefficients generated with the backstepping parameter configuration no-
DAF-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.10 (Experimental) comparison of horizontal inertial acceleration (Axb) produced by the
wing modulation at f = 2.5Hz and Vair = 5ms−1: (above) biological data of C.
brachyotis specimen; several measurements reported in (24), (below) BaTboT; several
estimations of Axb based on measurements of inertial thrust fxb. . . . . . . . . . . . . 169
10.1 Closed-loop environment with sliding-mode morphing control. Source: the
author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.2 Closed-loop environment with PID morphing control. Source: the author. . . 191
xx
List of Tables
2.1 Key Glossary of bat physiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Description of the 27 individuals used in the study. Source: (20). . . . . . . . . . . . 40
3.2 Scaling factors for wing morphological parameters as a function of body+wing mass
mt (cf. Figure 3.2). Source: (20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Scaling factors for wing aerodynamics parameters as a function of body’s mass mt (cf.
Figure 3.4). Source: (20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Detailed wing segment geometry at downstroke (segments according to Figure 3.7a.) . . 46
3.5 Key bio-inspired geometrical parameters for modeling and design. . . . . . . . . . . . 47
3.6 Biological-based framework for modeling and design. mt = 0.125Kg . . . . . . . . . . 47
4.1 Modified Denavit-Hartenberg parameters per wing. . . . . . . . . . . . . . . . . . . . 53
4.2 (Forward flight) coefficients of the third-order polynomial curves that describe Cartesian
paths of wrist and wingtip frames fitted to the xb, yb coordinates of the body frame. . . 58
4.3 Spatial operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Parameters for SMA phenomenological model . . . . . . . . . . . . . . . . . . . . . . 73
4.5 (forward flight) wing torques as a function of the wingbeat frequency f . . . . . . . . . 77
4.6 Characterization of actuators: wing torque requirements for flapping at 10Hz and
morphing1 at 2.5Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.7 Wing and body mass influence in the generation of τθ and τφ (mt = 0.125kg) . . . . . 85
4.8 Advantage of using bio-inspired wing joint trajectories on the production of pitching
and rolling torques). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.9 Limits for overloading SMA operation . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1 Main structural components of BaTboT. CAD design . . . . . . . . . . . . . . . . . . 94
5.2 Comparison of morphological parameters between the specimen and BaTboT . . . . . 95
xxi
LIST OF TABLES
5.3 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Circuit pinout data for SMA migamotor actuator. . . . . . . . . . . . . . . . . . . . . 100
5.5 General values of current consumption . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.6 Fabrication costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.1 Summary of SMA actuation performance. . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2 List of robot’s parameters: morphological, modeling, control. . . . . . . . . . . . . . . 144
7.1 Performance data of SMA actuation for longer periods of time. . . . . . . . . . . . . . 160
7.2 Lift and drag measurements for an angle of attack of 10o and Vair = 5m/s . . . . . . . 164
7.3 List of parameters used for experiments in figure 7.8. . . . . . . . . . . . . . . . . . . 165
7.4 Summary of performance of backstepping+DAF control and its influence into wing
modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
xxii
GLOSSARY
Nomenclature
a = distance from axis zi−1 to zi measured along xi−1, (m)
Ab = extended wing area, (m2)
AoA = angle of attack, (deg)
B = extended wing length, (m)
CL, CD = lift and drag coefficients
d = distance from axis xi−1 to xi measured along zi, (m)
f = wingbeat frequency, (Hz)
i = subscript that indicates the joint frame of the wings
fi = force of body i with respect to joint frame {i}, (N)
Fi = spatial force of body i with respect to joint frame {i}FL, FD = lift and drag forces measured about body frame {b}, (N)
Fnet = net forces measured about body frame {b}, (N)
FT = propagated spatial forces of the wings with respect to the base frame {0}H = projection onto the axis of motion
Ixx, Iyy , Izz = moments of inertia of body i with respect to frame {cm} (Kgm2)
Ii,cm = spatial inertia of body i with respect to frame {cm}Ib = spatial body inertia with respect to the body frame {b}Isma = electrical current input to SMA actuators, (A)
Ji,cm = inertial tensor of body i with respect to frame {cm}lh, lr = humerus and radius bones length, (m)
Li,cm = spatial inertial moment of body i with respect to frame {cm}Mb = body and wing mass, (Kg)
ni = torque of body i with respect to joint frame {i}, (Nm)
pi,i+1 = 3x1 position vector that joins frame {i} to {i+ 1}pi,i+1 = skew symmetric matrix corresponding to the vector cross product of pi,i+1
Pi,i+1 = spatial translation from joint frame {i} to {i+ 1}Psma = input heating power to SMA actuators, (W )
qi, qi, qi = joint positions, velocities and accelerations of body i with respect to {i} , (rad, rad/s, rad/s2)
ri+1,i = 3x3 basic rotation matrix that projects frame {i+ 1} onto frame {i}Ri+1,i = spatial rotation from joint frame {i+ 1} to {i}Rsma = electrical resistance of SMA actuators, (Ω)
si,cm = 3x1 position vector that joins frame {i} to {cm}si,cm = skew symmetric matrix corresponding to the vector cross product of si,cm
Si,cm = spatial translation from joint frame {i} to {cm}Ti+1,i = 4x4 homogeneous transformation matrix that relates frame {i+ 1} with {i}u = control inputs
U = 3x3 identity operator
υi, υi = linear velocity and acceleration of body i with respect to joint frame {i}, (m/s,m/s2)
Vi, Vi = spatial velocity and acceleration of body i with respect to joint frame {i}Vb = six-dimensional body accelerations with respect to the body frame {b}α = is the angle from axis zi−1 to zi measured about xi−1, (rad)
λ = backstepping+DAF control gains
φ, θ = roll and pitch angles measured about body-frame {b}, (rad)φd, θd = desired roll and pitch angular acceleration functions (DAF), (rad/s2)
τφ, τθ = rolling and pitching torques measured about body-frame {b}, (Nm)
ωi, ωi = angular velocity and acceleration of body i with respect to joint frame {i}, (rad/s, rad/s2)
xxiii
GLOSSARY
Acronyms
CAD Computer-Aided Design
CM Center of Mass
DAF Desired Angular acceleration Function
DC Direct Current
DH Denavit-Hartenberg
EoM Equations of Motion
FCA Flight Control Architecture
LiPo Lithium-Polymer
MAV Micro Aerial Vehicle
MCP MetaCarpoPhalangeal
PID Proportional-Integral-Derivative
PWM Pulse Width Modulation
SMA Shape Memory Alloy
SME Shape Memory Effect
UAV Unmanned Aerial Vehicle
VTOL Vertical Take-Off and Landing
xxiv
1
Introduction
”Flying. Whatever any other organism has been able to do man should surely be able to do
also, though he may go a different way about it.”
– Samuel Butler.
Bats exhibit extraordinary flight capabilities that arise by virtue of a variety of unique me-
chanical features. Bats have evolved with powerful muscles that provide themorphing capability
of the wing, i.e. folding and extension of the wing during flight. To change wing morphology,
bat wings are made of flexible bones that possess independently controllable joints (24), and a
highly anisotropic wing membrane containing tiny muscles that control the membrane tension
(27). This high degree of control over the changing shape of the wing has a great impact one
the maneuverability of the animal (28), (21), (29).
In recent years the concept of morphing Micro Air Vehicles (MAVs) has gained interest
(30), (31), (32), (17). The possibility of having actuated wings has allowed the design of
new mechanisms that improve over classical fixed/rotary-wings MAV flight performance. As
a result, different morphing-wing concepts and materials have emerged together with control
methodologies that allow for accurate wing-actuation (33), (34), (35), (36), (37), (38).
The concept of morphing-wings comes from nature (39), (40). Recently, the biological
community has demonstrated a special interest in understanding and quantifying bat flight
motivated by the sophistication of their flight apparatus (24), (21), (20). The wings are highly
articulated with independently controllable joints actuated by powerful muscles that provide
the animal with a high degree of control over the changing shape of the wing during flight.
In addition, tiny muscles embedded into the highly anisotropic wing membrane contribute in
controlling the membrane tension and camber (41). Bat wings also incorporate tiny hairs that
1
yo
xo
zo
{o}
{b}
yb
zb
xb
roll
pitch
yaw
humerus bone
radius bone
legelbow joint
wrist jointshoulder joint
wing membrane
MCP-III
IVV
τφ τθ
Figure 1.1: BaTboT. The overall mass of the skeleton+electronics+battery is 125g. The
wingspan: 53cm (wings fully extended). Each wing of the robot has six degrees of freedom
(dof): 2-dof at shoulder, 1-dof at elbow, and 3-dof at wrist joint. The body frame {b} is a 6-dof
floating body. Rotations about the body-frame {b}-xb, yb, zb axes are designated roll, pitch and
yaw following aerodynamic conventions. Frame {o} is the inertial frame.
sense airflow conditions, and there is some evidence that this sensing apparatus contributes to
their flight efficiency (8). There is no other flying creature in nature with a similar morphing-
wing system (28), (42).
Attempting to mimic the mechanics basis of bat flight seems to have great potential to
improve the maneuverability of current micro aerial vehicles. To closely mimic the morphing-
wing mechanism of bats, muscle-like actuation seems to be an adequate solution. In this regard,
Shape Memory Alloys (SMAs) have opened new alternatives with the potential for building
lighter and smaller smart actuation systems (43), (44), (45), (46). To the best of the authors’
knowledge, there is no morphing-wing MAVs in the state-of-the-art with highly articulated
wings inspired by the biomechanics of bats, actually the only works attempting to reproduce
bio-inspired bat flight using SMAs are presented in (10) and (47). Most of the experiments
in (10) were carried out with only a two degree of freedom wing capable of flapping at 3Hz.
Despite the fact that their robot is able to achieve accurate bio-inspired trajectories, the results
presented lack experimental evidence of aerodynamics measurements that might demonstrate
the viability of their proposed design. Moreover, neither (10) or (47) detail how to control the
SMAs for achieving the bio-inspired motion of the wings. Other works have also explored how
2
1.1 The problem and motivations
to optimize aircraft performance based on the aerodynamics of bat wings (11), (48).
Motivated by the potential behind bat flight and the lack of highly articulated morphing-
wing MAVs (not necessarily bat-like), this thesis presents a novel bat-like micro air vehicle
inspired by the morphing-wing mechanism of bats: BaTboT (cf. Figure 1.1). This thesis is
about:
The design and fabrication of the first highly articulated morphing-wing bat-like
robot. A novel strategy for the flight control will allow BaTboT to efficiently
maneuver by means of modulating wing inertia, without the need of any extra
mechanism such as ailerons, rudders, or back tails.
1.1 The problem and motivations
”Bats, the mysterious nocturnal mammals that are guided by sound, might hold the secret to
more-efficient flying machines.”
The problem
Morphing-wing aircrafts have emerged as a direction to enhance the efficiency of flight by
changing the wing profile. There is growing interest in the energy cost of flight and learning
from nature is the key to optimize efficiency. However, nature flyers such as insects, birds
or bats have extreme complexity in their flight apparatus and attempting to mimic part of
that complexity using artificial counterparts presents several challenges. Among these flyers
bats have evolved with truly extraordinary aerodynamic capabilities that enable them to fly
in dense swarms, to avoid obstacles, and to fly with such agility that they can catch prey
on the wing, maneuver through thick spaces and make high speed 180o degree turns. More
importantly, biologists have discovered that by flexing their wings inward to their bodies on the
upstroke, bats use only 65 percent of the inertial energy they would expend if they kept their
wings fully outstretched. Unlike insects and birds, bats have heavy muscular wings with hand-
like bendable joints and it is precisely this higher degree of dexterity that allows bats to save
energy during flight than any other flying creature.
The main problem to tackle in this thesis is how to optimize efficiency in terms
of net force production (inertial and aerodynamical) by developing a novel micro
3
1.1 The problem and motivations
aerial vehicle prototype with unprecedent highly articulated morphing-wings
inspired by the bat flight apparatus.
Solving this problem can help further the continued development in small unmanned flying
vehicles that waste minimum energy at the expense of incrementing payload capacity.
The hypothesis
In nature bats modulate wing inertia to improve on dynamics and aerodynamics response.
Based on this biological fact the following question is formulated:
Could a micro aerial vehicle inspired by the biomechanics of bats take advantage of the high
dexterity provided by the morphing-wings aimed at improving flight efficiency?
To this purpose, the following hypothesis is proposed:
Quantifying the effects of wing inertia in terms of thrust and lift production and
therefore including wing inertia information into the flight controller will allow
for the proper modulation of wing kinematics that finally would produce and
increase of net forces, thereby improving on flight efficiency.
Motivations: learning from bats
Bats can carry up to 50 percent of their weight and execute airborne maneuvers that would
make a bird or plane fall out of the sky. Bats use sophisticated echolocation to navigate, but
on top of that, hundreds of tiny hair sensors on the wing membrane that feed flight data are
used by the animal to improve on aerodynamics performance. Non other flying specie resemble
the way how bats sense airflow and adjust the wings to improve on fight.
Wing mass is important and it is normally not considered in flight. In bats, there is biological
evidence that the inertial forces produced by the wings have a significant contribution into the
attitude movements of the animal, even more significant than aerodynamic forces (24), (20).
In fact, bats perform complex aerial rotations by modulating solely wing inertia
(26), (23). This means bats are able to change the orientation of the body during
flight without relying on aerodynamic forces and instead by changing the mass
distribution of its body and wings. Inertial forces are likely to be significant in bats
4
1.1 The problem and motivations
a) b)
c) d)
VTOL
Figure 1.2: a) bats are agile flyer for hunting preys in air, water or even ground, b) bats can hover
like hummingbirds, consuming less energy (10 − 20Hz of flapping frequency), c) bat wings can
camber, stretch, extend and fold like no other flying animal in nature, d) VTOL flight capacity,
Source: Breuer Lab, http://brown.edu/Research/Breuer-Lab/research/batflight.html
because the mass of the wings comprises a significant portion of total body mass,
ranging from 11% to 33% and because wings undergo large accelerations (49).
Unlike birds or insects, whose wings are comparatively rigid and lighter, bats have wings
with more than two dozen independent joints, much like a human hand. This allows them to
manipulate the thin, flexible membrane that covers the wings in ways that can generate more
lift or greatly reduce drag. Surfaces of bat wings also curve more than a bird’s (camber) –
providing greater lift for less energy – while their extraordinary flexibility allows them to make
a 180-degree turn in less than half their wingspan, a radius impossible for any bird or existing
plane. Alike hummingbirds bats also can hover and power VTOL flight (vertical take-off and
landing) but with the main difference of saving energy due to the wingbeat frequency. Figure
1.2 shows some stills of amazing maneuvers bats are able to perform during flight.
The review of specialized literature reveals that many evolutionary biologist are attempting
to unlock the secrets of bat flight. Nonetheless, from a robotics perspective, lack of research
could be due to the complexity of the flight apparatus and the challenges to mimic part of that
complexity using an artificial counterpart.
5
1.1 The problem and motivations
Bats exhibit extraordinary flight abilities due to their unique wing structure which is quite
different then in other flyers like birds and insects. The flexible, flapping wings of bats may
pave the way toward versatile new types of aircraft, evolutionary biologist at Brown University
have revealed. 1 The sophisticated analysis looked at bats in a wind tunnel to uncover the key
differences between how mammals and birds stay aloft. Bats turn out to have a high degree of
control over the changing shape of their wings. These mammals can therefore generate lift as
their wings move both up and down; a big advantage when hovering.
Also aerodynamic forces generated by bat wings during flight are far more complicated
than those of birds. Bird wings operate almost as if they were airplane wings on hinges. By
comparison bat wings are more flexible and the material of the skin and bones are more stretchy
(the bones actually bend when the bat is flying). At slower speeds the morphing wings of bats
seem to have advantages in terms of maneuverability and energy savings, key issues that would
make bat-like aircrafts superior to bird-like or conventional aircrafts, especially in search-and-
rescue operations and covert surveillance.
To change wing morphology bat wings have an extremely high degree of articulation compris-
ing the elbow, wrist and finger joints which makes it more feasible to reproduce its mechanical
parameters using existing light materials. Its tendons and muscles are much smaller than those
found in birds and thus easier to model. With a thickness that varies in the range of 0.04 to
0.15mm, the wing membrane consists of many elastic veins and tiny muscle fibers that allow
the wing to be extended, folded and cambered. The skin of the membrane is actually very stiff,
its elasticity relying upon the fine and wrinkle texture that flattens out to create a taught airfoil
when the wing is extended.
The advent of the smart materials made possible the design of light wings that mimic the
complex motion of the morphing flight. E.g, the elbow and the wrist joints can be reproduced
such that the wing can fold and rotate its different segments minimizing the drag and the
negative trust in the upstroke motion. Artificial muscles designed from smart materials actuate
these motions during both downstroke and upstroke portions of the wingbeat cycle. By using
shape memory alloys (SMAs) as muscle like actuators behaving as biceps and triceps along the
wing-skeleton structure of the robot, the wings can extend and contract under the control of
the SMA wires that switch between two shapes when different currents are applied. The wires,
between the ”shoulder” and ”elbow” of the robot, rotate the elbow, pulling in the ”fingers” to
1Part of this research has been done with the collaboration of the Swartz Lab and Fluid Mechanics Laboratory
at Brown University.
6
1.2 Objectives
slim the wing profile on the upstroke. SMA might enable the contraction and extension motions
the wings in a similar way to the biological counterpart but more important, it can provide the
design of light actuated morphing-wings.
1.2 Objectives
This thesis presents a novel bat-like Micro Aerial Vehicle BaTboT with actuated morphing
wings that can be efficiently modulated by a novel flight controller that uses wing inertia
information to that purpose. The goal, to improve on flight performance in terms of lift/thrust
production and drag reduction.
In brief, this thesis provides both theoretical and experimental foundations for designing
highly articulated morphing-wing MAVs aimed at enhancing flight performance via proper wing
modulation (not necessarily bat-like).
The specific objectives of this research are:
1. To analyze and select which bat-specie would be suitable to be mimicked by an artificial
counterpart.
2. To study the mechanistic basis of bat flight. Based on published biological data that
unveils key aspects of bat’s morphology, physiology and aerodynamics performance, to
define a biological-based framework useful for designing a bio-inspired bat-like robot.
3. To formulate mathematical models for: i) body and wing kinematics, ii) dynamics (iner-
tial contribution), iii) aerodynamics (lift and drag production), and iv) SMA muscle-like
actuation.
4. To validate mathematical models against experimental data.
5. To design and fabricate BaTboT using the proposed biological-based framework.
6. To formulate a morphing-wing controller for the proper regulation of SMA actuators.
7. To formulate an attitude controller for the proper morphing-wing modulation that pro-
duces forward and turning flight.
8. To analyze and discuss the performance of BaTboT in terms of: i) accurate and fast SMA
actuation of morphing-wings, ii) inertial contribution on thrust production, iii) lift-to-drag
ratio, and iv) power consumption.
9. To demonstrate BaTboT would be capable of forward and turning flight via wind tunnel
testing without the need of external appendices such as rudders, ailerons, propellers, etc.
10. To discuss about the potential of the proposed methodologies towards real flight.
7
1.3 Methods
1. Biological study of bat flight 2. Bio-inspired design criteria
3. Methods: modeling + simulation
4. BaTboT fabrication 5. Control (measurements)
6. Experimental results
SMA actuation
dynamics(wing modulation)
aerodynamics
Figure 1.3: Structural steps to be followed during the thesis aimed at the development of BaT-
boT. The pictures depicted herein, correspond to the final BaTboT prototype. The forthcoming
chapters will introduce each step with all the details. Source: The author.
1.3 Methods
This subsection briefly summarizes the methods used for the development of BaTboT. The
following procedures will be approached aimed at achieving the main goals of this thesis. Figure
1.3 graphically details these procedures which will be all cover within each chapters of this
document.
1. Biological study of bat flight:
It presents a detailed study of the most relevant issues that describe bat flight: i) bio-
mechanics, ii) morphology, iii) physiology, iv) muscle actuation, v) kinematics, and vi)
aerodynamics performance. This study has been based on the most specialized biological
literature review from (50), (51), (52), (22), (23), (24), (20).
8
1.3 Methods
2. Bio-inspired design criteria:
It quantifies key design criteria based on the studied biological data analyzed in the
previous procedure. It summarizes these criteria into three fields: i) morphology, ii)
kinematics and iii) aerodynamics. Most relevant morphological parameters are: wingspan,
body and wing mass, wing area and wing-bone lengths. Kinematics parameters are:
wingbeat frequency and wingstroke trajectories. Aerodynamics parameters are: lift and
drag forces, wing membrane camber and angle of attack.
3. Modeling:
It defines morphology and kinematics frameworks of BaTboT. Wing kinematics are for-
mulated using modified Denavit-Hartenberg (DH) convention frames(53), whereas body
kinematics is designated by roll, pitch and yaw motions with respect to the body-frame
(following aerodynamic conventions (54)). Basic rotation matrix are used to express how
kinematics variables are propagated from the wings to the body. This allows for the
formulation of an integrated inertial model that mainly consists on: i) Newton-Euler
dynamics equations of motion expressed by spatial algebra notation (55), and ii) SMA
thermo-mechanical actuation based on existing phenomenological models that describe
the shape memory effect (56). Here, SMA performance is quantified in order to assess
the limits of this actuation technology. Also, the influence of wing inertia on robot’s
maneuverability is analyzed using the inertial model.
4. Design-Fabrication:
It approaches the design/fabrication problem. It shows a detailed description for the bio-
inspired development of: i) body and wing skeleton, ii) wing membrane, iii) actuation
mechanisms, and iv) hardware components.
5. Control (measurements):
It tackles the control problem. Two control layers are developed: i) morphing-wing
controller and ii) attitude controller. The former regulates the amount of input heating
power to be delivered to the SMA muscles. SMAs actuate to change the shape of the
wings (contraction/extension). The latter drives the former. It regulates the attitude
motion of the robot (roll y pitch) by means of proper wing modulation.
The novelty of the attitude control strategy is due to the incorporation of wing iner-
tia information within the control strategy. The idea behind this approach is aimed at
improving the attitude response of the bat-like MAV. The proposed controller is called
baskstepping+DAF (DAF, desired angular acceleration function). Such enhancement is
9
1.4 Original Contributions of this Work
based on the assumption (motivated by the cited biological studies) that bats efficiently
generate forward thrust by means of inertia wing modulation, taking advantage of relevant
wing-to-body mass ratio.
6. Experimental results:
It concludes with experiments aimed at:
• assessing the performance of the SMA muscles driven by the morphing-wing con-
trol. Performance will be quantified in terms of actuation speed, output torque and
fatigue,
• evaluating the accuracy of the attitude controller for tracking pitch and roll references
under the presence of external disturbances caused by aerodynamics loads at high
airspeeds,
• demonstrating the assumption of incrementing net body forces thanks to the wing
inertial modulation driven by the attitude controller, and
• showing the potential of the proposed methodologies toward achieving the first bat-
like MAV capable of autonomous high maneuverable flight.
1.4 Original Contributions of this Work
The original contributions of this work cover four areas:
Bio-inspired design and modeling
I BaTboT is the first Micro Aerial Vehicle composed by highly articulated wings, with 12
degrees of freedom counting both wings. No other platforms in the literature have a similar
amount of joints in their wings-system.
II The design process of BaTboT has been entirely conceived based on a comprehensive
analysis of biological data. The data from in-vivo experiments reported in the specialized
literature allowed for the definition of a bio-inspired design framework which defines every
aspect related to morphology, kinematics, and aerodynamics as a function of body and
wing mass.
III BaTboT is the first bio-inspired Micro Aerial Vehicle capable of maneuvering just by using
biomechanics parts found in nature, such as: wings, legs, the body, the wings’ membrane,
10
1.5 Thesis outline
etc. Most of the concepts found in the literature make use of extra mechanical parts that
are based on avionics principles, such as: ailerons, propellers, rubbers, back tails, etc.
IV An inertial model aimed at studying the influence of wing inertia on the production of net
forces for maneuvering.
SMA actuation and power
V Identified linear models for a ©Migamotor SMA actuator relating output torque with
input power.
VI Quantification of SMA limitations in terms of fatigue and actuation speed. It defines
the trade-off between input power, output torque, and actuation speed. This trade-off is
essential for the designing process of SMA muscle-like actuation mechanisms in similar
applications.
VII Accurate and faster position control of the SMAs (up to 2.5Hz in actuation speed) thanks
to re-adapted anti-overload and anti-slack mechanisms from (9). Normal rates of SMA
actuation speed range between 1 − 2Hz. It also uses SMAs as sensors, saving on weight
and energy.
Attitude control
VIII An enhanced backstepping control law denoted as backstepping+DAF. The Desired angular
Acceleration Function (DAF) incorporates wing inertia information aimed at the proper
modulation of the wings. This improves on attitude tracking and increments the production
of inertial thrust and reduces drag forces.
1.5 Thesis outline
Each chapter of this thesis beings with a General Overview about the problems to be addressed
and a brief description of the methods to be introduced. Thereby the end of each chapter con-
cludes with brief remarks about the topics presented. This document is organized as follows:
Chapter 2 is about a Literature Review. State-of-the-art research is introduced from
specialized literature covering areas such as: i) Nature flyers with a focus on bat biology, ii)
11
1.5 Thesis outline
Shape Memory Alloys as an alternative for actuation, and iii) Current Micro Aerial Robotic
platforms.
Chapter 3 is about biological inspiration. Key parameters of biomechanics basis of bat
flight are studied and unified into a bio-inspired framework for robot design. Relevant biological
data is also highlighted aimed at the proper formulation of robot’s models.
Chapter 4 is about Modeling. Based on biological data, BaTboT’s morphology, kine-
matics, dynamics, identified aerodynamics, and SMA actuation are defined and modeled using
mathematical frameworks. Basic maneuvers are defined by showing the influence of wing inertia
modulation on robot’s maneuverability.
Chapter 5 is about Design. Here, the design process and fabrication of BaTboT’s compo-
nents are introduced. It shows novel approaches for bio-inspired design and for the development
of low-mass high-power circuits.
Chapter 6 is about Control. It presents novel control techniques to: i) approach faster SMA
morphing-wing modulation and ii) enhance attitude regulation that allows for more efficient
flight control.
Chapter 7 presents the experimental tests carried out. Experiments are conducted to
demonstrate: i) morphing-wings control accuracy and speed, ii) SMA performance, iii) aerody-
namics performance, and iv) overall flight control.
Chapter 8 concludes the thesis with important remarks on the obtained results. Conclu-
sions are focused on the ares of: i) bio-inspired bat design, ii) SMA as muscle-like actuators,
iii) BaTboT’s overall control, and iv) General performance of the platform.
12
2
Literature Review
I may be biased, but I think bat flight is inherently fascinating.
2.1 General Overview
The knowledge from mechanisms of the biological species is the key input for designing intelli-
gent systems and exploring new technologies that allow to mimic key functionalities that nature
has perfected during millions of years of evolution (50), (24).
This chapter presents a comprehensive review on key information related to the physiologi-
cal and morphological functions of biological bats, aimed at providing an integrated view of the
structural design of a bat-like robot. A specialized literature survey from (57), (58), (51), (59),
(60), (61), and (1) will provide significant insights into the requirements to mimic key function-
alities of biological flight, with an special emphasis on attempting to replicate the mechanistic
basis of the bat flight apparatus using smart materials (62), (43), concretely, Shape Memory
Alloys (SMA) acting as artificial muscles that can be controlled to provide high dexterity of
wing modulation (morphing-wings).
This chapter reviews the state-of-the-art in relation to:
• Nature flyers: comparison of some aspects of bat biomechanics with those from birds and
insects.
• Bat flight research: biological insights on quantifying bat flight.
• Shape Memory Alloys (SMA) for muscle-like actuation.
• Morphing-wing MAVs that use smart actuators.
13
2.2 Nature flyers
2.2 Nature flyers
The variety of flying animals is staggering, ranging from mosquitoes to hummingbirds, birds,
flying foxes, eagles. And yet they all rely on the same physical processes and almost the same
mechanism of locomotion. Nonetheless, key variations of that mechanism through evolution
has defined different patterns of flight, specialized according to the environment of each flying
specie. This section highlights some of the difference between nature flyers, emphasizing into
the unique mechanism of the bat flight apparatus.
2.2.1 Biomechanics: insects, birds, and bats
Four groups of animals have evolved powered flight: i) insects, ii) pterosaurs, iii) birds, and
iv) bats. Though these groups evolved powered flight independently, all use roughly the same
flapping pattern (cf. Figure 2.1). The wings of the three vertebrate groups (pterosaurs, birds
and bats) all evolved from a modified forelimb. Pterosaurs, extinct relatives of dinosaurs, had
wings supported mostly for a single, enormously elongated fourth finger and a wing membrane
stretched from the arm/hand skeleton to the side of the body.
Butterfly Draganfly
Cockroach Beetle
FlyTrue bug
Bats
Birds
pterosaurs
Figure 2.1: Structure of animal wings showing the main skeletal support. (Left) vertebrates,
(Right) insects.
Among living vertebrates, bats are the only mammals able to fly. Their wings are su-
perficially similar to Pterosaurs wings in that they have greatly elongated fingers supporting
a flexible membrane, but their anatomy is actually quite different mainly because the arm
skeleton (upper arm and forearm) supports much larger proportion of the wing in bats than
14
2.2 Nature flyers
Pterosaurs. In addition bats can adjust its four wing fingers independently, gaining more level
of flight control.
On the other hand, birds have a different wing-system, mainly because most of the surface
of their wings consists of feathers. The arm/hand skeleton extends barely half the length of the
wind, and the fingers are reduced to little more than three short spurs of bone.
Insects were the first animal group to evolve flight, and they are the only invertebrates to
have done so. Insects wings are fundamentally different compared to vertebrates in several
ways. Vertebrate wings are all modified legs (forelimbs), while insect wings evolved separately
from their legs. Insect legs are attached to the bottom of the thorax and the wings are attached
to the upper side of the thorax. Flying vertebrates have muscles as well as bones with joints
out in the wing, so vertebrates flyers can directly control the shape and movements of their
wings. This functionality of changing wing shape is also known as morphing wings. In insects,
the membrane contains embedded tiny veins that carry blood vessels. These veins are anchored
to a complex set of tiny skeletal structures, -axillary scleritis-, that make up the wing hinge. By
adjusting the position of these structures, the insects can push or pull on different veins, which
can, in turn, adjust the shape of the wing. However, due to their anatomy, that kind of wing
morphology is limited and they cannot flex their wings or bend them to shorten their span.
In that regard, bat morphing wings are more sophisticated, with an incredible potential for
high maneuverability flying at lower speeds compared to birds. To facilitate the choice of an
optimal type of natural wing as a model for the BaTboT design, a number of small natural
fliers are thoroughly analyzed from specialized literature, and their morphological and flight
parameters are compared particularly in terms of the wingspan, wingbeat frequency, the wing
loading, lift-drag, and power.
Wingspan (S) and Aspect Ratio (AR) The wingspan (cf. Figure 2.2a) is perhaps the
most important morphological measurement required on a flyer, after body mass. It is the
distance between the two wing tips when the wing is fully extended during the downstroke:
S = 2lm + 2B, (2.1)
where 2lm is the body width of the flyer and B is the mean wing length. A longer wingspan
is generally more efficient because the flyer suffers less induced drag and its wingtip vortices
do not affect the wing as much. However, the long wings mean that the flyer has a greater
moment of inertia about its longitudinal axis and therefore cannot roll as quickly and is less
15
2.2 Nature flyers
a) b)
d)c)
Figure 2.2: Comparison for several species of small-scale flyers: a) Wingspan, b) Aspect ratio,
c) Wing loading, d) Wingbeat frequency. Source from (1), (2), (3).
maneuverable. The aspect ratio AR (cf. Figure 2.2b) is a powerful indicator of the general
performance of a wing. It describes the shape of the wing by means of the ratio between the
wingspan and the wing’s mean chord, as defined in Eq. (2.2). High aspect ratio wings are
associated with a reduced power cost to create comparable lift. The species that use a gliding
flight, like condors, eagles exhibit a high aspect ratio that allows them to stay aloft for large
periods of time using the least amount of energy.
AR = S2A−1b , (2.2)
Where S is the wingspan and Ab the mean wing area. The damselflies tend to have the
highest aspect ratio between the studied species, making them suitable for gliding flight with a
reduced amount of energy lost through wingtip vortices. In general, bats present relatively low
aspect ratios indicating a higher maneuverability with a loss of performance due to a higher
induced drag.
Wing Loading
Wing loading (WL) refers to the ratio of the body mass to the wing surface area of the
flyer. This measurement relates the flight speed with the flyer’s maneuverability. Lower wing
16
2.2 Nature flyers
loading allows slower flight (still producing necessary lift), whereas larger wing loading makes
the animal to fly faster.
WL =MbA−1b , (2.3)
Maneuverability is also dependent on wing loading since the minimum radius of turn is
proportional to body mass. Comparing the wing loading in Figure 2.2c, (24) has shown that
bats are able to achieve turns of 180deg in just three wingbeats using an intricate combination
of banked and crabbed mechanisms. By studying the wing kinematics during these turns, (23)
has shown that bats can maneuver in narrow spaces less than half of their wingspan.
Wingbeat Frequency (f) is one of the most interesting flight parameter for flapping flight
and is affected by the body mass, wingspan, wing area and the wing moment of inertia. Using
a combination of multiple regressions and a dimensional analysis, (63) experimentally derived
that the larger the wingspan S, the smaller is the wingbeat frequency f to keep the flyer aloft,
and that an increase of the body mass, should be balanced by an increase wingspan S and
in wing area Ab in order to reduce the wingbeat frequency. Figure 2.2c shows the average
comparison of wingbeat frequencies for several flyer species.
Lift, Drag, and Reynolds number
Lift is defined as the component of force orthogonal to flow and so perpendicular to drag
(cf. Figure 2.3a). The lift and drag forces of the wing are defined relative to the direction of
the airflow over the wing, not relative to the directions in which the animal flies or gravity acts.
As a rule, lift is perpendicular and drag is parallel to the direction of the airflow. Lift and drag
have a resultant force, which is tilted forward during the downstroke and because of this it has
a vertical component, the upward force, and a horizontal component that is tilted forward, and
so it is providing the thrust.
The total lift force FL can be expressed through definition of a lift coefficient CL. Likewise,
the total drag FD acting on a flier is the sum of the effects of pressure and viscous drag. The
comparison of drag of different fliers is facilitated through the definition of a non-dimensional
drag coefficient CD. Both lift and drag are calculated as:
FL = 0.5CLρAbV2air
FD = 0.5CDρAbV2air
(2.4)
17
2.2 Nature flyers
a) b)
c) d)
Figure 2.3: a) Cartoon of aerodynamics forces acting on a typical wing design, b) Lift and drag
changes with the angle of attack for a typical wing design. c) Comparison of the lift to drag ratio.
d) Power Flight. Source from (1), (2), (3).
The term Ab is the mean wing area of the flyer and Vair os the airspeed. Both lift and drag
forces acting together with inertial forces are related by the Reynolds number Re. In other
words, the formulation of the Reynolds number is given by the ratio of inertial of inertial to
viscous forces varies with the ratio of the fluid density and the fluid viscosity (2). Most bat
specimens range from a Re between 103 and 104. Also, another important characteristic of a
given airfoil is the lift to drag ratio, L/D (cf. Figure 2.3c). The higher the lift to drag ratio
of the airfoil, the less thrust is needed to produce that required lift. Furthermore, bats use 35
percent less energy by reducing aerodynamic drag (cf. power-to-mass ratio in Figure 2.3d).
2.2.2 Bat biology
Bats are mammals, which means that they are warm-blooded, have fur, and produce milk. Bats
belong to the Order Chiroptera within the class Mammalia of the kingdom Animalia. Within
the Order Chiroptera bats can be divided into two groups depending on its taxa: Microchi-
roptera and Megachiroptera. Microchiroptera (or microbats) in general navigate by echolocation
generating ultrasound via the larynx that emits the sound through the nose or the open mouth.
Microbats call range in frequency from 14k to over 100k hertz, well beyond the range of the
human ear (typical human hearing range is considered to be from 20Hz to 20kHz). Microbats
18
2.2 Nature flyers
are 4 to 16cm long. They usually have short faces, well-developed tails, and do not have a claw
on their second finger. Megachiroptera (or megabats) are ”fruit-bats” that generally navigate
by sight (although a few use echolocation). In contrast to Microchiroptera, these bats have a
longer face, and a claw on their second finger. The largest (Pteropus vampyrus) reach 40cm
in length and attain a wingspan of 182cm, weighing in at nearly 1kg-f. The smallest bat is
the Craseonycteris thonglongyai, an insect-eating bat that has a wingspan of only 0.15m and
weighs few gram-force.
Glossary and terminology
Table 2.1: Key Glossary of bat physiology
Physiology Terminology
Membrane Skin
Patagium: is the wing membrane having an average thickness of about 0.03mm. It is
composed by elastic fibers and bundles of muscle fibers.
Plagiopatagium: is the large portion of the wing, between body and fifth digit. This is
the lift-generating section and its camber is controlled by flexing the body axis and/or the
fifth digit.
Propatagium: is the small portion between the shoulder, elbow and Carpus. It is also
used specially to produce lift.
Dactylopatagiums: is the digit membrane and is especially a propelling or thrust gen-
erating portion. Its camber can be changed by flexing the digits.
Main Bones
Scapula: provides attachment for many of the flight muscles and is mobile across the back
of the rib cage during the wing beat cycle
Clavicle: enables a wider arc of rotation for the humerus. During the upstroke the scapula
slides back to its more dorsal position.
Humerus: bone that connects the shoulder and the elbow joints.
Radius: bone that connects the elbow and the Carpus joints.
Third MetaCarpoPhalangeal (MCP-III): digit bone that connects the wrist/carpus
joint with the wingtip point.
Main joints
Shoulder: Complex 3-DoF joint formed by three bones: scapula, clavicle and humerus.
It is useful for producing flapping motion.
Elbow: 2-DoF joint useful for achieving wing morphing capacity. 1-DoF produces the
forearm (radius) flexion and the second one that produces the roll of the wrist resulting in
a leading edge flap.
Wrist/Carpus: Complex 3-DoF joint capable of achieving roll, pitch and yaw motion
(similarly to humans wrist). Roll helps in generating more thrust, pitch helps in flexing
the palm to approach the forearm in a downward and upward direction, and yaw helps in
flexing the forearm in a backward direction.
Main muscles
Pectoralis major, subscapularis, serratus anterior, clavodeltoideus: These are the
main four muscles involved during the wing downstroke cycle.
Trapezius, costo-spino-scapular, deltoid: These are the main three muscles involved
during the wing upstroke cycle.
Supraspinatus, triceps, biceps, occipito-pollicalis, coraco-cutaneus, humeropata-
gialis, tensor plagiopatagii: These are the group of muscles that control the flight
membranes.
Since a large part of terminology has a biological provenience, it is useful (from an engineer-
ing point of view) to introduce and explain some of the most frequent terms used for describing
bat physiology (64). These terms are consigned in Table 2.1. The forthcoming subsections will
19
2.2 Nature flyers
Figure 2.4: Bat anatomy. A) body anatomy, B) Pectoral skeleton, C) Dorsal view of retracted
right arm, D) Dorsal view of expanded right arm. Source (4).
graphically show and explore bat physiology in more detail.
Bat anatomy
In terms of anatomy bat wings are built on the basic pattern of a mammalian limb. It is
an analogous structure to the human arm and hand (in fact, the name ”chiroptera” is Greek
for ”handwing”), but the relative sizes of most bones and muscles are very different. The bat
also has unique muscles in the patagium, chest and back, to power the wing during flight.
The wing consists of the upper arm, forearm, wrist and hand. The bones of the hand and
the four fingers are greatly elongated, light and slender to provide support and manipulate
the wing membrane, called the patagium (cf. Figure 2.4). The wings of bats are much thin-
ner than those of birds, so bats can maneuver more quickly and more accurately than birds (63).
Bat flight muscles
Bats use the pectoral muscle as the main adductor or depressor, which means that draws the
wing toward the median line of the body (52). In (65) Prof. Norberg studied the direction of pull
of the main flight muscles in bats, given an insight about how wings move during downstroke
and upstroke cycles of the wingbeat.
20
2.2 Nature flyers
Pectoralis Subscapularis
Serratusventralis
Depressors (Adductors)
Levators (Abductors)
Rhomboideus
Spinotrapezius
Spinodeltoideus
ClavodeltoideusSupraspinatusAcromiotrapeziusAcromiodeltoideus
Downstroke UpstrokeM. subscapularis
ScapulaM. trapezius
M. rhomboideusM. spinodeltoideus
M. acromiodeltoideus
Clavicle
SternumM. pectoralis
M. serratus
M. clavo-deltoideus
BAT stroke muscles
Figure 2.5: Bat muscle structure to power wingstroke motion.
As shown in cf. Figure 2.5, bats have four downstroke muscles: pectoralis major, subscapu-
laris, part of serratus anterior, and part of the deltoid . Together, these muscles constitute
about 12% of the bat’s body weight. The upstroke is powered when required by the remainder
of deltoid, trapezius, the rhomboids, infraspinatus, and supraspinatus (costo-spino-scapular).
The scapula provides attachment for many of the flight muscles and is mobile across the back
of the rib cage during the wingbeat cycle.
The first digit of the wings is small and clawed and it is used during climbing and walking.
The muscle extensor carpi-radialis muscle inserts on the base of the metacarpal of the second
digit. This muscle pulls forward on the metacarpal and a ligament between digits II and III
transfers tension to the third digit, thus keeping the outer part of the wing taut and extended.
Digits IV and V extend across the chord of the wing. The muscle abductor along the ventral
surface of the 5th digit can bring about changes in the camber of the membrane (cf. Figure
2.5).
In this thesis, bats’ physiology, specially the muscles, are fundamental for the understand-
ing of maneuverability and efficiency during flight. In addition, wing kinematics will be also
derived from the analysis of how muscles move the wing bone structure and membrane in order
to generate the wing stroke patterns.
21
2.2 Nature flyers
M coraco - cutaneusPropatagium
M occipito - pollicalisM humeropatagialis
Chiropatagium
Plagiopatagium
M bicepsM triceps
M tensor plagiopatagiiM extensor carpi radialis
M extensor carpi ulnaris
Embedded tiny muscleswithin the membrane
Figure 2.6: Bat wing membrane
Wing Membrane
The proximal segment of the wing (plagiopatagium and propatagium) produces most of the
lift, and the distal segment, chiropatagium, produces most of the thrust during the wingbeat
cycle (cf. Figure 2.6). The proximal segment is maintained at an appropriate camber and angle
of attack for the production of lift throughout the wingbeat cycle. The degree of curvature of
the fifth digit has an important role in determining and maintaining the angle of attack and
the camber of this segment. By retaining its curvature and angle of attack throughout the wing
beat cycle, the fifth digit partly controls the lift produced by the proximal segment (58).
When the bat is at rest its wings are folded in an accordion shape-like. Then, at the
beginning of the flight, a rapid extension of the fingers of the wings occurs, extension that is
generated by the contraction of a single muscle: the supraspinatus.
The opening speed of bat wings is very rapid due to its unique folding and unfolding patterns
(morphing-wing capability). This rapid folding and unfolding stems from the microstructure of
the bat wings. Swartz et al. (27) studied in details the microstructure of bat wing membrane
skin. Figure 2.6 shows overall structure of a bat wing with reinforcing frame (shoulder bone,
five fingers and hind limb). The membrane structure is a network of fibers which are spun
orthogonally and support thin skin. This corrugated skin structure helps easy folding and
unfolding actuations. The skin structure along the vertical direction exhibits high stiffness while
22
2.3 Bat flight research
that along the horizontal direction has higher failure strain. The high stiffness of the wing along
the vertical direction is needed to keep the wing shape. Also, the surface of the membrane
is equipped with touch-sensitive receptors on small bumps called Merkel cells, found in most
mammals including humans, similarly found on our finger tips (66). These sensitive areas are
different in bats as each bump has a tiny hair in the center, making it even more sensitive and
allowing the bat to detect and collect information about the air flowing over its wings, thereby
providing feedback to the bat to change its shape of its wing to fly more efficiently.
2.3 Bat flight research
Evolutionary biologists are deeply interested in understanding and quantifying all aspects of
bat flight in terms of aerodynamics, navigation, behavior and so on. In (5) a framework for 3D
reconstruction and analysis of bat flight maneuvers is presented. The reconstructed model of
the bat is composed by 52 degrees of freedom, giving a great insight about how bats modulate
the articulations of the whole body and wings. Figure 2.7 shows the tracking of bat flight in
real time.
Figure 2.7: Tracking results of a bat using a 52 degree of freedom articulated model. Shown on
top are frames extracted from high speed video of a landing bat. Shown on the bottom are the
corresponding frames of the reconstructed three-dimensional wing and body kinematics. –Caption
extracted from (5)–.
Using this model in conjunction with the measured wing kinematics, (5) has shown that
surprisingly modulation of wing inertia plays the dominant role of reorienting the bat with little
or no reliance on aerodynamic forces.
Thus, bats are able to change body orientation mid-flight without relying on
aerodynamic forces and instead by changing the mass distribution of its body and
wings.
23
2.3 Bat flight research
Figure 2.8: Reconstruction of the wake of C.
brachyotis bat. Source: (6).
Another remarkable research regarding
the relationship of bat flight kinematics and
wake structure is presented in (6). Wind-
tunnel measurements of the bat specie
Cynopterus brachyotis have revealed the
resultant wake velocities produced by the
kinematics motions of the wings. Figure
2.8 shows that a closed loop vortex struc-
ture is dominant at relatively slow forward
flight speeds (4.3m/s), and there is evi-
dence for additional small vortex structures
shed from other appendage. Besides at-
tempting to quantify the kinematics of bat
flight and wake vorticity, the wing mem-
brane is
a material with key features that make bat flight unique among nature. In (7). in-flight wing
membrane strain measurements have been carried out. Data were collected from wind tunnel
wind-off flights of a Jamaican fruit bat, Artibeus jamaicensis, is shown in Figure 2.9.
Results have shown that strain levels are around 10% in the X direction and 3% in the Y
direction. Surface snapshots of the membrane strain-state show in the X direction (spanwise)
a consistent strain-relief effect around the ring finger during downstroke, with high-strain con-
centration areas on the membrane between the little finger and the ring finger. The estimates
of the shape and positions of two section of the hand wing during the downstroke
were evaluated and revealed that in order to achieve this strain distribution, the
membrane should have a mid downstroke camber value that is a function of the
body mass of the specimen Mb. This relationship has been defined as: wing camber
∝ M0.9b . Depending on the specimen’s size, this value typically varies from 0.07 to
0.25, as measured in (20).
Continuing with the membrane, the material of what it is made is not the whole thing.
There is biological evidence that bats have tiny hairs in the membrane that sense airflow
conditions, and there is some evidence that this sensing apparatus in bats contributes to their
flight efficiency (cf. Figure 2.10). In (8), evidence that the tactile receptors associated with
24
2.4 Shape Memory Alloys
Figure 2.9: Strain measurements of bat’s
wing membrane. Source: (7).
Figure 2.10: Sensory wing hair before (A)
and after (B and C) depilation. (A) Scanning
electron microscope image from a domed hair
located on the ventral trailing edge (location
is marked by a gray circle in schematic to the
Right) of Eptesicus fuscus. Caption extracted
from Source: (8).
these hairs are involved in sensorimotor flight control by providing aerodynamic feedback. It was
found that neurons in bat primary somatosensory cortex respond with directional sensitivity
to stimulation of the wing hairs with low-speed airflow. Wing hairs mostly preferred reversed
airflow, which occurs under flight conditions when the airflow separates and vortices form. This
finding suggests that the hairs act as an array of sensors to monitor flight speed
and/or airflow conditions that indicate stall .
2.4 Shape Memory Alloys
Within the need of building smaller devices with integrated features of sensing and actuation,
the field of smart materials have opened a new generation of actuation devices (43), (62). The
word ”smart” has been used to highlight the property of some materials capable of changing
its physical properties upon certain conditions. Some of these materials are based on: i) Piezo-
electricity, which react to the application of an electric field, ii) Shape Memory Effect (SME),
which possess the ability to actuate when subjected to thermal changes and recover its initial
configuration without any thermal process involved, iii) Electro Active Polymers (EAPs), either
ionic or electronic, which directly exploit Maxwell forces or the electrostrictive phenomenon to
25
2.4 Shape Memory Alloys
obtain mechanical energy from electrical input energy, iv) Electrorheological fluid (ERF), whose
rheological characteristics vary, depending on the external fields applied, and v) Magnetostric-
tion, either positive or negative, where magnetic domains are reoriented by means of an external
magnetic field.
This thesis involves the use of Shape Memory Alloys (SMAs) as both actuators and sensors
of wing motions. Basically, the SMAs make use of the Shape Memory Effect, exhibiting a
thermally activated martesitic transformation. The development and application of SMAs
attained the requisite momentum following the discovery of NiTi alloys in 1963 at the Naval
Ordnance Laboratory. The name nitinol refers to Nickel, Titanium, NOL (Naval Ordnance
Laboratory). The implementation of applications making use of SMAs has evolved hand-in-
hand with the development of nitinol. In the last few decades, both industry and academia have
evinced growing interest in the application of nitinol, and of SMAs in general. This is basically
because these materials are intrinsically susceptible of use both as sensors and as actuators,
which makes them suitable for use as smart actuators and for integration in smart structures.
This section provides a concise description about the use of SMAs as actuators in robotics,
by describing how the material operates and some of the applications in terms of modeling and
control. Also, state-of-the-art Micro Aerial robots that use smart materials for morphing-wing
actuation are highlighted from the literature.
2.4.1 Basic foundations
Fist Principles
In SMAs, the shape memory mechanism is based on a reversible, solid-state phase transforma-
tion between the high temperature austenite phase and the low temperature martensite phase.
This phase transition is also known as martensitic transformation. The martensitic phase trans-
formations of the alloy can be characterised by four transformation temperatures: i) As, the
austenite start temperature, ii) Af , the austenite finish temperature, iii) Ms, the martensite
start temperature, and iv) Mf , the martensite finish temperature. These transformations are
shown within the hysteresis curve of the SMA in Figure 2.11a.
When the temperature is less thanMf , the NiTi alloy consists only of the martensite phase.
As the temperature is increased beyond As, austenite begins to form in the alloy and when
the temperature exceeds Af , the alloy is primarily in the austenite phase. As the alloy cools,
martensite begins to form when the temperature drops below Ms, and when the temperature
26
2.4 Shape Memory Alloys
(a) (b)
(c) (d)
Figure 2.11: a) The hysteresis curve of SMA, b) stress-strain-temperature curve of SMA ex-
hibiting the one-way shape memory effect, c) stress-strain-temperature curve of SMA exhibiting
the two-way shape memory effect, d) stress-strain-temperature curve of SMA exhibiting pseudoe-
lasticity behavior. Source: (9).
reaches Mf , the alloy is again fully martensitic. As mentioned, these transitions represent the
SMA thermal hysteresis loop.
During phase transitions between martensite and austenite, most of the physical properties
of SMAs vary. These include Young’s Modulus, electrical resistance, heat capacity and thermal
conductivity (9), (67). Considerable research for modeling the microscopic and macroscopic
behavior of SMAs has been carried out in the last decades. Since the mechanical behavior is
closely related to microscopic phase transformations, stress-strain relations are not applicable
to describe the shape memory effect.
A single SMA wire describes one-way shape memory effect when the alloy shows permanent
deformation after the removal of an external force. Therefore, it can recover its original shape
upon heating. Subsequent cooling does not change the shape unless it is stressed again. In
addition, a two-way shape memory effect may occur upon cooling and without the applying of
external stress. This phenomenon can be easily observed within an antagonistic arrangement
of SMA wires acting in parallel. Figure 2.11b depicts the one-way shape memory effect. By
heating the deformed martensite past As, the austenite start temperature, austenite begins to
27
2.4 Shape Memory Alloys
form and the material begins to contract. Full shape recovery can be achieved by heating above
Af , where the alloy is completely in the austenite phase again. As this shape recovery only
occurs in one direction, it is referred to as the one-way shape memory effect.
On the other hand, the two-way shape memory effect (as described in Figure 2.11c) can
be defined as the reversible shape change upon thermal cycling in the temperature range of
martensitic transformations without requiring any external load. SMAs can be trained to
exhibit the two-way effect using two methods, which are spontaneous and external load-assisted
induction (68), (69). However, the shape change obtained is in practice less than the one-way
effect.
Another interesting phenomenon is the pseudoelasticity, which is the shape recovery associ-
ated with mechanical loading and unloading of SMAs at temperatures above Af . Figure 2.11d
depicts stress-strain temperature curve associated with the pseudoelasticity behavior of a 2D
crystal structure model of SMAs. There is no temperature change required for pseudoelastic
behaviour. Therefore, the strain characteristic can be described using only the stress-strain
plane (9).
Modeling
There are two classes of SMA models: microscopic and macroscopic. The former is based on
first principles models, whereas the latter is based on experimental results provided by system
identification. In this thesis, both models will be approached, the former for simulation purposes
and the latter for control design.
The microscopic model uses in this thesis has been presented by Elahinia in (56) and (70).
The model is described as a multi-dimensional thermomechanical constitutive phenomenological
model. The model is based on Tanaka’s pioneer work in (71). Details on this microscopic model
can be found in Section 4.5.2. Similar phenomenological models can be found in the literature:
Kuribayashis model based on experimentally identified relations (72), (73), which observed a
linear relationship between very small variations in the force and strain of an SMA wire. Under
constant strain, the relationship between the force and supplied voltage was also observed to be
approximately linear. The sublayer models of Hirose et al. (74), (75) introduced the two-phase
model for SMAs using a commonly method in solid mechanics to describe nonlinear stress-
strain relationships. The model considers the SMA to be composed of parallel sublayers of the
different phases with their respective mechanical properties. This is combined with a model of
transformation kinetics based on thermodynamics to form the variable sub-layer model. Laterm
28
2.4 Shape Memory Alloys
Madill et al., in (76) extended their work for a new SMA actuator model that is capable of
modeling minor hysteresis loops. Others, such as Liang and Rogers (77), Brinson (78) and
Elahinia (70) improved upon Tanaka’s model using different transformation kinetic equations
relating the martensite fraction to the stress and temperature.
To approach macroscopic modeling based on system identification process, this thesis uses
the procedure described in (9) which consists on small-signal response of SMA wires over a
suitable frequency range aimed at defining the relationship between input power and the output
force of the wire during contraction. As demonstrated by Yee in (9), Kuribayashi in (73), and
Grant (79), the power-to-force relationship of the AC response of a NiTi SMA wire can be
modeled similar to a first-order transfer function. Section 6.3 will detail on this issue.
Actuation and Control
Basically SMAs are used as two modes of actuation: one actuator composed of a SMA wire and
a bias spring, or two SMA wires featuring an antagonistic configuration. Herein we have used
the latter, also called, differential SMA actuator. In 1984, Honma et al., (80) demonstrated
that it is possible to control SMA actuation by electrical heating. They used open loop Pulse
Width Modulation (PWM) to operate the actuation. The first control of an antagonistic pair
of SMA wires for moving a rotational joint was presented by Kuribayashi in (72). PWM was
used in a feedback controller to control both position and force. The control scheme adjusted
the duty cycle between the two antagonistic actuators by switching the applied voltage between
the two actuators.
Currently SMA control can be divided in three categories: i) Linear control, ii) PWM,
and iii) non-linear control. Linear control is widely extended, featuring PID methodology for
the control strategy. Some of the more notable work include (81), (82), (83), (84), and more
recently, (9) and (67). On the other hand, PWM technique has been implemented by some of
the early SMA researchers including (80), (82), (72), and more recently, (85).
Non-linear control of SMAs is also widely extended. Pons et al., in (58). compared PI control
based on direct strain feedback linearisation and feedforward approach to the conventional PI
controller. It was shown that feedforward control achieved the best overshoot reduction. Other
comparisons with linear controllers include Lee and Lee [43] who investigated time delay control
on SMA actuators, and Ahn and Nguyen [1] who experimented with self-tuning fuzzy PID
controllers.
29
2.4 Shape Memory Alloys
In terms of improving the actuation speed of SMAs, Yee et al., (9), (67) studied different
phenomena on NiTi SMA wires, from small-signal high frequency response analysis, to force
models based on system identification, and the proper mechanisms that allow for faster and
accurate force control of an antagonistic pair of SMA actuators. These mechanisms are called:
i) anti-slack, and ii) anti-overload. The former deals with the two-way shape memory effect (62),
improving accuracy and speed, whereas the latter limits the amount of input heating power
to prevent physical damage when SMAs are overloaded. As a result, their force controller
was capable of tracking fast and accurate force references compared to other works from the
literature (86), (79), (87). Nonetheless, their control architecture requires of a high-bandwidth
force sensors capable of providing the force feed-back.
In this thesis, a control architecture similar to the one described in (67) has been imple-
mented, which makes use of both anti-slack and anti-overload mechanisms to manage the afore-
mentioned limitations in SMA actuation speed and accuracy. However, some important changes
to the architecture are proposed, since the available payload capacity of the bat robot constrains
the use of force sensors. Instead, SMA electrical resistance feed-back is used for sensing mo-
tion (88), (89). Thereby, both anti-slack and anti-overload mechanisms have been conceived for
regulating the amount of input heating power based on a Resistance-Motion (RM) relationship.
This technique will be detail on Section 6.4, while experimental results in Section 6.5 will show
proper position control performance in terms of tracking and actuation speed of the SMAs,
applied to the morphing-wing mechanism of the bat robot.
2.4.2 Advantages and drawbacks
Most relevant advantages and drawbacks of using SMA technology for actuation are highlighted
as follows:
Advantages
• Size and weight : SMAs can be directly used as linear actuators. There is no need for
additional motion components or hardware, which permits easy miniaturizations of the
actuation system. For the application at hand, BaTboT requires the minimum added
weight in order to be able to fly. Other actuation mechanisms, such a DC-motors, servos,
etc, are simply inappropriate due to their weight. Instead, SMA wires have a negligible
volume (e.g., 3× 10−9m3), allowing for extremely light wings.
30
2.4 Shape Memory Alloys
• High Force-to-weight ratio: SMA actuators have a large force-weight ratio of ∼ 8N/1 ×10−5Kg, using a wire with thickness of 150μm, and 0.1m long. SMAs also present large
life cycles (3× 106).
• Noise-free operation: Because SMA actuators do not require friction mechanisms such as
reduction gear, it avoids the production of dust particles, sparks and noise. These merits
make SMA actuators extremely suitable for areas such as microelectronics, biotechnology
and biomimetics applications (high bio-compatibility).
• Sensing properties: Although SMAs are mostly used for actuation, they also have sensing
capabilities. Several properties of the SMAs change as it undergoes martensite phase
transformation. Among these properties is the resistivity that decreases as the tempera-
ture of the wire increases and hence its phase transforms to austenite. A liner relationship
between electrical resistance change and SMA strain can be derived.
Drawbacks; challenges to tackle
• Slow speed : SMA actuators have generally been considered to have slow response due to
restrictions in heating and cooling, and also due to the inherent thermal hysteresis. The
common method in actuation is by electrical heating. Although applying larger electrical
currents can increase the speed, this may also overheat and damage the actuator without
monitoring. Most research so far has investigated SMA position control at generally low
tracking speeds of less than 1Hz. Rise times for step responses usually took more than 1
second.
• Fatigue: Long-term performance of the Shape Memory effect could decrease over time if
the material is expose to large external stress or overheating temperatures resulting from
large input currents.
• Low energy efficiency : The maximum theoretical efficiency of SMAs is of the order of 10%
based on the Carnot cycle, according to (90). In reality, the efficiency is often less than 1%,
since the SMA actuator can be considered a heat engine operating at low temperatures.
This means that the conversion of heat into mechanical work is very inefficient. Most of
the heat energy is lost to the environment. Hence SMA actuator applications must be
limited to areas where energy efficiency is not an issue (cf. (9)).
31
2.5 Morphing-wing MAVs with smart actuation
a)
b)
Figure 2.12: a) The BATMAV robot, b) Detailed arm assembly using SMAs-bases muscles.
Source: (10).
2.5 Morphing-wing MAVs with smart actuation
This Section reviews the state-of-the-art Micro Aerial Robots (MAVs) that use smart materials
for actuation, more concretely Shape Memory Alloys, although not limited to this material.
Other platforms featuring smart materials, such as Piezoelectric or polymers are also included,
which are not necessarily biologically inspired.
BATMAV
To the best of the authors’ knowledge, the only works attempting to reproduce bio-inspired
bat flight using SMAs are presented in (10) and (47). A robotic platform called BATMAV
(fully actuated by SMA wires) is described in both papers. Thereby, SMAs have been used for
two purposes: first, as muscle-like actuators that provide the flapping and morphing wingbeat
motions of the bat robot, and second, as super-elastic flexible hinges that join the wings’ bone
structure. The BATMAV prototype is shown Figure 2.12.
32
2.5 Morphing-wing MAVs with smart actuation
a)
b)
c)
d)
Figure 2.13: a) Bat-wing platform in the wind-tunnel, b) Nylon membrane, c) Spandex mem-
brane, d) Silicone membrane. Source: (11).
Most of the experiments in (10) were carried out with a two degree of freedom wings capable
of flapping at 3Hz. Nonetheless and despite the robot is able to achieve similar bio-inspired
trajectories, the results lack of experimental evidence relating aerodynamics measurements that
demonstrate the viability of the proposed design. Moreover, neither (10) or (47) detail how to
control the SMAs for achieving the bio-inspired motion of BATMAV’s wings.
Bat-Wing
The research presented in (11) is aimed at the study of bat wing biomechanics and aerody-
namics performance for the design of an artificial wing. The wing prototype is composed by a
basic skeletal structure based upon relative dimensions from the anatomy of a bat. Different
wing models were assembled comparing different materials for the membrane, such as: Nylon,
Spandex, and Silicone. Wind tunnel measurement have been carried out to analyze the wing
performance in terms of Lift-to-drag production. Figure 2.13 shows the Bat-wing prototype.
The results have shown that that silicone is simply a more aerodynamically-suited material.
This material is non-porous, is able bend along with a flexible skeleton to create optimal flight
shapes, and appears to be able to withstand contact with winds up to 20mph to create lift
forces.
Another interesting research from the same authors (48) has also studied key features of bat
flight in order to understand how certain aspects of the wings might improve on the design for
micro air vehicles. The goal of the study is not to mimic natural bat flight, but to understand
33
2.5 Morphing-wing MAVs with smart actuation
veins
moldedpolymer
carbonfiber
piezo-motor
bi-morphingtransmission
(thorax)
power electronics
Figure 2.14: Harvard RoboBee, actuated by PZT. Source: (12).
how certain aspects of bat flight apply to the engineering problem of wing design for micro air
vehicles. Aspects such as morphing, cambering, and twisting are measured and quantified in
terms of lift production and drag reduction/
Robobee
Harvard’s scientist in (12) present an innovative insect-scale robotic thorax designs capable of
producing asymmetric wing kinematics similar to those observed in nature and utilized by flies
and other two-winged insects to maneuver. Inspired by the thoracic mechanics of such insects,
a Piezoelectric (PZT) actuator has been fabricated as a bending bimorph cantilever actuator.
The transmission maps the approximately linear motions of the actuators into the flapping
motion of the wings. The transmission consists of links and joints with geometries designed to
maximize the product of stroke amplitude and first resonant frequency, given known actuator
and airfoil properties. The insect-like robot, shown in Figure 2.14 is capable of flapping at 90Hz.
Actuators are created by laminating two piezoelectric plates (PZT-5H from Piezo Systems,
Inc.) to a carbon fiber spacer and electrode layer. The airframe and transmission are created
by layering a 7.5μm polyimide film (Chemplex Industries) with carbon fiber face sheets.
Morphing trailing edge
A trailing edge being developed by Wildschek et al., in (13) splits the trailing edge into seamless
top and bottom control surfaces that are morphed independently. The inner structure consists
of hollow segmented parts separated by bars and hinges that allow for both compliance and
34
2.5 Morphing-wing MAVs with smart actuation
Figure 2.15: Inner structure and possible control modes of a multi-functional trailing edge.
Source: (13).
aerodynamic loading. The model has been built out of carbon reinforced plastic that is mor-
phed through the use of an all- electric actuation system. The model is capable of multiple
actuation modes (pitch, roll, high lift, etc). Within the airfoil, SMAs have been used in a spring
configuration for providing the wing actuation. The wing prototype is shown in Figure 2.15
and could be used for improving fixed wing current aircrafts’ design.
Morphing-wing camber MAV
Wilkie et al., in (14) have developed a MAV capable of varying the camber of its wings. Varying
the camber in a wing can have beneficial properties for the control of an aircraft such as during
take-off and landing when the lift distribution along a wing is required to dramatically change.
The MAV is depicted in Figure 2.16, which consists of a macro-fiber composite (MFC); a
flexible film with a layer of unidirectional piezoceramic fibers sandwiched between layers of
copper electrodes and acrylic/Kapton. Wind tunnel tests revealed improved drag characteristics
due to having a continuous wing surface as opposed to discrete articulated control surfaces, a
characteristic shared by many morphing structures. Asymmetric and symmetric actuation of
the MFC patches provides roll and pitch moments of around 0.06Nm.
Other morphing-wing MAV concepts
In general, most of the morphing-wing MAV concepts out there relies on standard motor ac-
tuation technology. Some of the best concepts are provided by FESTO smart-bird (cf. Figure
2.17a) (15), where its wings not only beat up and down, but also twist at specific angles. This
is made possible by an active articulated torsional drive unit, which in combination with a
complex control system attains an unprecedented level of efficiency in flight operation. Festo
35
2.5 Morphing-wing MAVs with smart actuation
Figure 2.16: Layout of an MFC actuation device for MAV morphing-wing camber. Source: (14).
Figure 2.17: TOP (left to right) Smart-bird by FESTO, inspired by the Seagles (15), Combat by
University of Michigan, inspired by bat’s navigation system (16), DARPA nano-air hummingbird
(http://www.darpa.mil). BELOW (left to right) The Gull Wing by (17), MFX-1 by NextGen
Aeronautics, inspired by the batwing internal structure (18), Prototype of Entomoter MAV (19).
has thus succeeded for the first time in creating an energy-efficient technical adaptation of this
model from nature.
University of Michigan researchers have came out with a bat-llike MAV concept, shown in
Figure 2.17b (16). The project is funded by DARPA, aimed at the development of a robotic
bat with the ability to navigate at night, using low-power miniaturized radar and a very sen-
sitive navigation system. The robotic bat will also have the ability to navigate at night, using
low-power miniaturized radar and a very sensitive navigation system. Its lithium battery will
recharge using solar energy, wind, and vibrations, and the bat will communicate with the troops
using radio signals. Bats have a highly-attuned echolocation sense providing high-resolution
navigation and sensing ability even in the dark, just as our sensor must be able to do. Echolo-
cation allows bats to navigate by emitting sounds and detecting the echoes. The robot’s body
concept is designed to be about six inches long and to weigh about a quarter of a pound. Its
expected energy consumption will be 1W .
Lind et al., in (17) have developed and built a series of UAV models that incorporate
36
2.6 Remarks
a folding wing concept called a Gull Wing (cf. Figure 2.17c) The wings contain a telescoping
spar connected to a hinged spar that enables the folding motion. Another MAV that takes some
inspiration by bats’ wings design is presented in (18) (cf. Figure 2.17d). The MAV is called
the NextGen batwing, developed by NextGen Aeronautics. The MAV successfully underwent
a 40% planform area morphing, 30% wingspan morphing, and 20o sweep angle morphing in
mid-flight. Further studies have been made on optimization of the batwings cell structure and
actuator placement, as well as developing control laws to efficiently control the wings morphing.
2.6 Remarks
Bats have enormous advantages in both inertial and aerodynamics compared to other flying
animals. Their high body-to-wing has ratio and high wing dexterity allow bats to perform aggres-
sive maneuvers modulating solely wing inertial. Also, changing the wing profile improves of the
generation of lift forces and reduction of drag during the wingstroke. These factors might be the
key for developing micro aerial systems that attempt to mimmic efficient flapping/morphing
wing flight. From the biological review presented in this chapter one can immediately note why
the bat apparatus is worthy to mimic using BaTboT:
• Low wingbeat frequencies : as shown in Figure 2.2d, bats can produce enough lifting forces
by flapping their wings in the range of ∼ 10Hz. To achieve that, bats have larger
wingspans (Figure 2.2a) and low body mass. This combination is essential into the proper
production of lift. Note e.g., in Figure 2.2d, that Hummingbirds in the same body mass
scale compared to bats, require to flap their wings at least five times faster than bats. In
terms of design, the scale values for fabricating a robot with similar wingspan and body
mass are perfectly achievable with BaTboT.
• Morphing wings and cambering : bat wings are unique in nature. They contain powerful
muscles that provide the same dexterity and mobility (degrees of freedom) than the human
arms. It means they can fold, expand and camber their wings in such a unique way.
Furthermore, the wing membrane also contains embedded tiny muscles and veins that
contribute in controlling the tension of the membrane during flight (cf. Figure 2.6).
• Sensing : as mammals, bats are actually able to see very well. By combining great sight
and echolocation, they can maneuver in narrow spaces, during daylight or night. In
addition, bat’s wings have tiny hairs over the membrane’s surface that can sense the
airflow and adjust wing morphology to improve on aerodynamics.
37
2.6 Remarks
• Efficient flight : By combining both morphology and sensing capabilities, bats are the best
flying animals to maneuver at low speeds and altitudes than any biological mechanism
in nature. They can hover like a hummingbird, maneuver like birds, or change direction
abruptly like insects. On top of that, bats use 35 percent less energy by reducing aero-
dynamic drag, having the lowest power-to-mass ratio (W/g) of any flying vertebrate in
nature (cf. Figure 2.3d).
In terms of actuation, Shape Memory Alloys (SMAs) enable the fabrication of lighter wins
with muscle-like actuation but some challenges should be addressed. Section 2.4.2 highlighted
the advantages and drawbacks of this material. This thesis will present feasible solutions to
minimize the effects of SMA limitations and it will give an insight into the performance of the
material acting as actuators. The goal is not only to evaluate the use of this actuation technology
for the application at hand but also on providing a formal quantification of performance that
would allow others to drive this technology forward.
By reviewing the state of the art in Section 2.5, one can note the lack of biologically-inspired
robots that explore alternative actuation mechanisms more likely to those found in nature. The
field of bio-inspired MAVs that use smart materials for actuation is still in an early stage. Most
of the works have investigated how to fabricate efficient wings models, but few have achieved
to develop a complete bio-MAV platform capable of sustained flight. This thesis embarks into
this potential field by presenting the first highly articulated bat-like MAV that can maneuver by
means of changing wing morphology and also it takes advantage of the improvements in flight
efficiency that wing modulation enables.
38
3
From bats to BaTboT:
Mimicking biology
3.1 General overview
This chapter presents insights of in-vivo bat flight. BaTboT’s morphology and biomechanics
are based on the bat specie Rousettus aegyptiacus physiology. This section describes why the
selection of this specie to be mimicked with BaTboT. This selection has been based on criteria
regarding: i) wing morphology (i.e., wingspan, aspect ratio, body and wing mass, etc), ii) wing
kinematics (wing joint trajectories), and iii) wing aerodynamics (wingbeat frequency, angle of
attack, lift production, etc). The biological data is analyzed from in-vivo biological experiments
of several species carried out in (22), (23), (24), (20).
3.2 Review on biological flight performance data
Body size governs almost every aspect of animal biology. In bats, body size and mass, wing
mass and wingspan determine most of the kinematics, dynamics and aerodynamics performance
of the animal. Hence, it is reasonable to think that a proper choice of the specimen to mimic
should be based on the analysis of these parameters. (20) presents a rigorous study that
compares the wing kinematics of 27 bats representing six pteropodid species ranging more than
40 times in body mass (0.0278−1.152kg). This section briefly summarizes the results presented
in (20), relating and highlighting performance issues mostly in terms of wing kinematics and
aerodynamics.
39
3.2 Review on biological flight performance data
1
2
3
zo
xo{o}
wingtip
wristc d
ef
g
h
kjilm
n
qp
ob a
Figure 3.1: Right lateral view of the bat with respect to the inertial coordinate system {o}.Green dots are the path of the wrist joint whereas red dots are the path of the wingtip over a
wingbeat cycle. The position and posture of the right wing are shown at three time points in the
wingbeat cycle. Source: (20).
The biological experiments consigned in (20) consisted in recording the flight kinematics of
27 animals from six species shown in Table 3.1. The positions of 17 anatomical markers placed
on the individual’s wings were digitized via high-speed cameras. Five flights of the individuals
within the wind-tunnel were used for analyses. The bats were recorded performing forward flight
and describing the wing-trajectory profile shown in Figure 3.1.
Table 3.1: Description of the 27 individuals used in the study. Source: (20).
Species name Abbreviation (Color code) Body mass of specimens [kg]
Cynopterus brachyotis Cb(Purple) 0.028, 0.031, 0.035, 0.035, 0.040
Rousettus aegyptiacus Ra(Blue) 0.112, 0.132, 0.159
Pteropus pumilus Miller Pp(Green) 0.178, 0.178, 0.180, 0.204, 0.212
Eidolon helvum Eh(Yellow) 0.254, 0.266, 0.278, 0.326, 0.332
Pteropus hypomelanus Ph(Orange) 0.454, 0.464, 0.468, 0.490, 0.526
Pteropus vampyrus Pv(Red) 1.020, 1.052, 1.090, 1.152
3.2.1 Measurements of wing morphological parameters
The following parameters were calculated regarding how the specimens from Table 3.1 changed
their wings during the trials (cf. Section 2.2 for detailed definition of morphological parameters):
40
3.2 Review on biological flight performance data
Figure 3.2: (A) Maximum wingspan, (B) minimum wingspan, (C) wing chord, (D) maximum
wing area, (E) wing loading, and (F) aspect ratio. Circles represent medians for each species and
the black arrow points to the specimen under analysis. Source: (20).
• Maximum and minimum wingspan (bmax, bmin): bmax is two times the maximum distance
of the wingtip marker from the mid-sagittal plane yo = 0 (measured during downstroke).
bmin is the opposite (measured during upstroke).
• Maximum wing chord (cmax): the longest two-dimensional distance between the wrist
and the tip of digit V (e and n in Figure 3.1).
• Maximum wing area (Smax): the left wing was divided into 18 triangular surfaces (see
Figure 3.1). The areas of those triangles were summed, then multiplied by two, to arrive
at total wing area (S).
• Wing loading (Qs) and Aspect ratio (AR): defined as Qs = mtgS−1max, being mt the bat’s
mass, g is the gravity, and Smax the maximum wing area. AR = b2maxS−1max.
Figure 3.2 compares how morphological parameters among the specimens are affected de-
pending on the body mass mt. This study allows for the quantification of scaling factors that
determine relationship between morphological parameters and body+wing mass. Table 3.2
details these values.
3.2.2 Measurements of kinematics parameters
Similarly, kinematics parameters are shown in Figure 3.3. The horizontal flight speed scales
as: V ∝ m0.005t . The horizontal velocities of bats (4.98 ± 0.09ms−1) were much greater than
41
3.2 Review on biological flight performance data
Table 3.2: Scaling factors for wing morphological parameters as a function of body+wing mass
mt (cf. Figure 3.2). Source: (20).
Parameter Scaling factor
Minimum wingspan (bmin) bmax ∝ m0.423t
Maximum wingspan (bmax) bmin ∝ m0.366t
Maximum wing chord (cmax) cmax ∝ m0.357t
Maximum wing area (Smax) Smax ∝ m1.32t
Wing loading (Qs) Qs ∝ m0.233t
Aspect ratio (AR) AR ∝ m0.072t
Figure 3.3: (A) Flight speed, (B) Horizontal accelerations, (C) Vertical accelerations. Circles
represent medians for each species and the black arrow points to the specimen under analysis.
Source: (20).
vertical velocities (0.12± 0.03ms−1), so flight paths were close to horizontal plane (1.36± 0.36o
above horizontal).
3.2.3 Measurements of Aerodynamics parameters
Figure 3.4 compares the most relevant parameters that indirectly affects the aerodynamics of
the bats. Scaling factors are detailed in Table 3.3.
Table 3.3: Scaling factors for wing aerodynamics parameters as a function of body’s mass mt
(cf. Figure 3.4). Source: (20).
Parameter Scaling factor
Minimum wingbeat period (T ) T ∝ m0.180t
Downstroke duration (Tdown) Tdown ∝ m0.213t
Downstroke ratio (τdown) τdown ∝ m0.036t
Maximum wing stroke amplitude (φs) φs ∝ log(mt)(−3.058)
Strouhal number (St) St ∝ m−0.088t
Minimum angle of attack (α) α ∝ log(mt)(−7.738)
Wing camber at maximum span camber ∝ m0.9t
Minimum lift coefficient (CL) CL ∝ m0.170t
42
3.3 Choice of species
Figure 3.4: Wingbeat period (A) scaled lower than expected under isometry. Downstroke dura-
tion (B), Downstroke ratio (C), stroke amplitude (D), stroke plane angle (E) and Strouhal number
(F) did not change significantly with body mass. Angle of attack increased with body size (G) as
a result of a change in α1 (H), but not from a change in α2 (I). Wing camber (J) did not change
with body size, but coefficient of lift (K) did. Circles represent medians for each species and the
black arrow points to the specimen under analysis. Source: (20).
3.3 Choice of species
From the starting point of this thesis, the design process of BatBot was visualized and thought
within the range of bio-inspired Micro Air Vehicles (MAVs). Unfortunately there is no standards
or normative that strictly classify the most relevant properties of bio-MAVs. However, the
literature commonly use wingspan and mass properties as criteria for classifying different types
43
3.3 Choice of species
Birds
UAVs
Aircraft
Bio-MAV design scale
Insects
mas
s (K
g)
wingspan (m)10.10.01 10 100
1
0.1
0.001
100
10000
Bats
BaTboT
Figure 3.5: Bioinspired MAV classification
depending on wingspan and mass. Source:
the author.
Figure 3.6: Wing physiology. 17
markers are placed on: anterior
and posterior sternum (a and b, re-
spectively), shoulder (c), elbow (d),
wrist (e), the metacarpophalangeal
and interphalangeal joints and tips
of digits III (f, g, h), IV (i, j, k), and
V (l, m, n), the hip (o), knee (p),
and foot (q). Source: (21).
of Unmanned Air Vehicles (UAVs). Figure 3.5 details this classification.
In nature, bat’s wingspan varies from 0.15m (Craseonycteris thonglongyai micro bat) to
1.82m (Pteropus vampyrus mega bat). Likewise, the mass of each specimen varies from 8g to
1kg respectively. From the specimens considered in (20) whose study was briefly summarized
in the previous section, wingspan varies from 0.3m (Cynopterus brachyotis) to 1.1m (Pteropus
vampyrus Linnaeus), whereas mass varies from 0.028Kg to 1.15Kg respectively.
The selection of the specimen to mimic is based on four main criteria: i) efficiency
in lift production at ii) lowest flapping frequency with iii) about half-meter wingspan and iv)
minimum weight.
The Rousettus aegyptiacus also known as the Egyptian fruit bat fulfills with these criteria
(cf. blue-color dot in Figures 3.2, 3.3, 3.4). Another specimes with similar morphology and flight
performance: Pteropus poliocephalu and Cynopterus brachyotis are also taken into account into
the design process of BaTboT. The Rousettus aegyptiacus has an average maximum wingspan of
∼ 0.5m when the wings are fully extended (cf. Figure 3.2A) and a minimum wingspan of ∼ 0.2m
when the wings are folded (cf. Figure 3.2B). Wingspan size allows for a feasible fabrication of
BatBoT’s biomechanics using standard Fused Deposition Modeling (FDM) rapid-prototyping
in 3D. Also, this specimen has an average mass of ∼ 125g (cf. Table 3.1) which allows for a
feasible weight to obtain with standard Acrylonitrile Butadiene Styrene (ABS) material.
Besides morphology, the most important characteristic of this specimen is related to its
lift production. Normally, large bats have higher coefficients of lift than small bats. This is
obviously due to large wingspan and area. However, the Rousettus aegyptiacus is capable of
44
3.3 Choice of species
shoulder
wrist
MCPIIIwingtip
1 2 3 4 5 6 7 8 9 10 11 12 13 14
ep
0.049m 0.057
B = 0.197m (extended, downstroke)
0.090m
0.014m
WristIII
IV
V
(a) (b)
Wingtip
a
b
c
d
ef
g MCPIII
15deg
~15deg
f =0.032
h=0.014
g=0.012e=0.045
35.4deg63.3degreference
line
y
x
reference line
III
IV
V
wing joints
Figure 3.7: Detailed parameters that describe wing segment morphology. a) wing segment
subdivision, b) detailed configuration of the wrist joint and attached digits when the wing is fully
extended. It shows the angles between digits that maintain proper wing membrane tension during
downstroke. Source: the author.
efficiently produce lift forces at low wingbeat frequencies even with a mid-size wingspan. Note
in Figure 3.4K, the lift coefficient raises up to ∼ CL = 1.15 with the bat flapping at f = 6.6Hz
(cf. Figure 3.4A, wingbeat period T = 0.15s). Comparing the Rousettus aegyptiacus against
the Cynopterus brachyotis, note the former doubles CL by flapping about 50% slower compared
to the latter.
In the following, key aspects of the specimen morphology are determined as an useful frame-
work for BaTboT modeling and design.
3.3.1 Wing morphology
Key components of wing physiology are shown in Figure 3.6. In (22), 17 markers placed
along the wing allowed the quantification of wing size and proportions. Using these markers
the wing planform is divided into segments, allowing for the morphological description of the
wing using three geometrical parameters: i) wing chord, ii) leading edge position (b), and iii)
position of the center of mass (c). Figure 3.7 details these segments with measurements for the
specimen at hand. Also, Table 3.4 shows the values of geometrical parameters that describe
wing morphology.
Measurements of leading edge position (b) and the position of the center of mass (c) are
chordwise relative to a reference line through the left and right glenohumeral joint; negative
values are below the reference line. Figure 3.7a shows the detailed wing segments, and the
parameters that completely describe the size and proportions of the wing. Five key-dots (red
45
3.3 Choice of species
Table 3.4: Detailed wing segment geometry at downstroke (segments according to Figure 3.7a.)
Segment Chord a[m] Leading-edge position b[m] CM position c[m]
1 0.11 -0.035 0.004
2 0.107 -0.0347 0.0035
3 0.098 -0.0294 0.005
4 0.087 -0.0081 0.093
5 0.082 0.0086 0.014
6 0.083 0.0148 0.0198
7 0.086 0.032 0.03
8 0.09 0.036 0.032
9 0.092 0.0276 0.032
10 0.079 0.030 0.038
11 0.061 0.032 0.044
12 0.051 0.028 0.037
13 0.028 0.029 0.032
14 0.01 0.033 0.029
colored), corresponding to segments: #1, 4, 7, 11, and 14, are placed at the i) shoulder, ii)
elbow, iii) wrist, iv) MCP-III medium point, and iv) wingtip. The shoulder joint is connected
with the elbow joint through the humerus bone and the elbow is connected with the wrist
joint through the radius bone. Digits III, IV and V are connected to the wrist. The wrist
joint plays an important role into the kinematic of bat flight. Bats have very complex wrists,
similar to human hands. It allows for the rotation of digits in three-dimensions. Attempting
to mimic this complexity using an artificial counterpart is unfeasible mainly because the added
weight of required actuators to drive each digit. To simplify on this, only planar motion of
the digits is considered. This allows the digits to open and close by rotating about the gravity
axis aimed at maintaining the proper wing membrane tension during the downtroke. Figure
3.7b details the wrist and digits configuration. When the wings are fully extended, digits must
be completely open to keep the membrane with the largest tension. This improves on lift
production. To this purpose Figure 3.7b shows the maximum angles that separate one digit
from another (angles measured with respect to the joint frame of the wrist and the reference
line). The angles are calculated from key length-proportions defined at each segments. These
proportions are h = 0.014m which corresponds to the distance of each segment subdivision,
f = 0.032m which is the leading-edge position measured at segment #11 (parameter b in Table
3.4) and g = 0.012m which corresponds to the distance g = c− b at segment #11.
3.3.2 Biological-based framework for modeling and design
Concretely, Table 3.5 shows the most relevant geometrical parameters to be used for the mod-
eling and design of the robot. Also, Table 3.6 provides quantification about how morphology,
46
3.4 Remarks
Table 3.5: Key bio-inspired geometrical parameters for modeling and design.
Parameter [unit] Value
Total mass mt [g] 125
Extended wing length B [m] 0.195
Body width lm [m] 0.07
Extended wing span S = lm + 2B [m] 0.53
Extended wing area Ab [m2] 0.05
Humerus length lh [m] 0.055
Humerus average diameter 2rh [m] 0.0055
Radius length lr [m] 0.070
Radius average diameter 2rr [m] 0.0042
Plagiopatagium skin thickness [m] 0.0001
Table 3.6: Biological-based framework for modeling and design. mt = 0.125Kg
Parameters relation-value
mass/wing-length ∼ 0.35g/cm
Morphological Minimum wingspan m0.423t = 0.41m
Maximum wingspan m0.366t = 0.46m
Wing area m1.32t = 0.064m2
Minimum wingbeat period m0.18t = 0.68s
Kinematics Maximum wingstroke amplitude log(mt)(−3.05) = 157.8o
Minimum angle of attack log(mt)(−7.738) = 6.98o
Minimum lift coefficient m0.170t = 0.7
Aerodynamics Maximum wing camber1 m0.9t = 0.15
1 Please refer to (20) for a detailed calculation of the wing camber parameter.
kinematics and aerodynamics parameters should be defined taking into account the body+wing
mass of the robot mt. Both tables provide the biological-based framework for modeling
and design. Modeling is covered in the following Chapter 4.
3.4 Remarks
This chapter has allowed the understanding of biological parameters that directly affect bat
flight and provides the foundations and criteria for robot design. Analyses of biological ex-
periments described in (20) allowed for a complete definition of a set of key issues to consider
during the designing process of BaTboT. These issues show how morphology, kinematics and
aerodynamics can be related to each other into a bio-inspired designing framework, cf. Table
3.6 . The following chapter introduces the mathematical formulation for kinematics, dynamics,
aerodynamics and wing-actuation using SMA-like muscles.
47
4
BaTboT modeling
4.1 General overview
This chapter presents the modeling of the most important components involved within the de-
sign process of BaTboT: i) kinematics, ii) dynamics, iii) aerodynamics and iv) SMA for wing
muscle-like actuation.
The content of the following sections is summarized as follows:
• Kinematics: it defines the morphological parameters of BaTboT, empathizing into the
topology of the highly-articulated wings. This topology consists on two serial chains
(each wing) symmetrically connected to a floating base system (the body). Here, methods
for describing bio-inspired joint trajectories that determine the motion of the wings are
presented. The wing trajectories are aimed at producing forward and turning flight. This
requires an inertial model that will be presented in the dynamics section.
• Dynamics: it presents the inertial model. Newton Euler formalism is used to describe
dynamics Equations of Motion (EoM) of the entire bat-robot. Spatial algebra is used for
the formulation of the EoM. Spatial vectors (55) are those which the linear and angular
aspects of rigid-body motion are combined into a unified set of quantities and equations.
The notation to refer a spatial operator is X ∈ �6, being X any 6D vector. The advan-
tages of using spatial formulation is noted when writing computer code algorithms, which
makes the codes easier to read, write and debug. Using the inertial model allows the
quantification of inertial effects (forces) on thrust production and robot maneuverability.
48
4.2 Kinematics model
• Aerodynamics: It presents experimental quantification of lift and drag forces. It also
determines how to account for aerodynamics effects by incorporating aero terms into the
inertial model. It shows how to obtain the net forces of the system by considering both
inertial and aerodynamics contribution.
• SMA wing actuation : it defines a phenomenological model based on thermo-mechanical
equations that describe one-way shape memory effect behavior. This model is used for
the quantification of SMA performance in terms of actuation speed, output torque and
fatigue. This quantification also allows for defining the limits of SMA actuation for the
application at hand. The phenomenological model has been adapted from Elahinia’s
works in (56), (70).
Using the SimMechanics toolbox of ©Matlab, kinematics, dynamics and SMA actuation
models are integrated into a single module that represents the BaTboT’s plant. The advantage
of using SimMechanics relies on the possibility of importing all the mechanical properties from
the SolidWorks CAD model of the robot1. Simulations will be carried out aimed at determining:
• Torque requirements for wing actuation : the required torques to actuate the wing
joints are determined by the inertial model (solution of the inverse dynamics problem).
It allows for the characterization of actuators.
• Maneuverability : requirements for forward and turning flight are determined by the
inertial model. It quantifies and analyzes the influence of wing inertia on the production
of body accelerations.
• SMA limitations: the maximum allowed input electrical current that achieves the
fastest contraction and extension of the SMA actuators is determined by the phenomeno-
logical model. It allows to explore the limits to safety overload the response of the SMAs.
4.2 Kinematics model
In this section the kinematics framework of BaTboT is formulated. This framework is described
by three issues:
1. Topology. It describes the system as two chains of rigid bodies serially connected (each
wing) that are symmetrically joined to a base (body). The body is assumed to be a
1Further details about the CAD design can be found in Chapter 4.
49
4.2 Kinematics model
free-floating base that moves in �6. Wing frames denoted as {i} being i the wing joint
frame are placed for each degree of freedom of the wing joints.
2. Wing and body kinematics. It uses the geometrical parameters described in Table 3.5
to represent the morphology of the wings and the body. Modified Denavit-Hartenberg
(DH) convention is used to place wing frames of references, whereas Euler angles are
used to describe how the body frame rotates by following aerodynamic conventions. In
addition, spatial operators for rotation and translation based on 3x3 basic rotation matrix
are formulated. This allows for the solution of the forward kinematics problem and also
for the proper propagation of physical quantities from the wings to the body.
3. Wing trajectories. Joint trajectories for each wing are denoted by the term qi, being
the subscript i the wing joint frame. Joint trajectories have been directly extracted from
biological experiments carried out in (20), (22), (23), (24). Depending on wing modulation
joint trajectories allow for the generation of forward and turning flight. Here, simulations
to achieve both maneuvers are presented and discussed
4.2.1 Topology
Bat morphology is dimensionally complex due to the highly articulated wings. Bats have dozens
of wing joints and movement is influenced by the flexibility of the bone elements, the orientation-
dependent compliance of the membrane, their interactions with the surrounding fluid, and by
movements of the numerous tendons and muscles within the membrane. Attempting to mimic
part of that complexity is a challenge that requires identification of the most relevant joints and
its role in providing proper wing modulation. Despite bat wings are highly articulated, only six
joints are key for partially changing the wing shape. This makes the wing membrane skeletally
maneuverable by a jointed i) legs, ii) shoulder, iii) elbow, iv) wrist, and v) five fingers (digits)
connected to the wrist. Figure 4.1 shows wing and body topology.
Under this topology frames {1}-{6}R and {1}-{6}L represent the right and left wing respec-
tively, frames {b} and {0} are the body frames and frame {o} is the inertial coordinate system.
The following section details the convention to place these frames within the topology of the
robot.
4.2.2 Wing and body kinematics
Kinematics frames are described in Figure 4.2. Morphological and geometrical parameters are
defined in Tables 3.5 and 4.1.
50
4.2 Kinematics model
{b,0}{1,2}L
{3}L
{4,5,6}L
wingTip wingTip{1,2}R
{3}R
{4,5,6}R
{7}L {7}R
6-dof floating body {o}
zo
yoxo
body-frame {b} - 6dof floating body
frames {1},{2} - 2dof shoulder jointframe {3} - elbow jointframe {4,5,6} - 3dof wrist joint
frame {0} - base
{o} - Inertial frame
Figure 4.1: Topology. The robot has an overall of 14-DoF (not counting the 6-DoF of the floating
body). Each wing has 6-DoF and each leg 1-DoF. Source: the author.
Body-frame
The body frame {b} has xb pointing cranially along the body axis, yb pointing laterally toward
the right wing, and zb points downward and lies in the plane of symmetry of the body. The
rotation of a rigid body in space can be parameterized using several methods: Euler angles,
Quaternions, Tait-Bryan angles, etc. The most extensively used method in aerospace engi-
neering is the Euler angles, which consist in a mathematical representation of three successive
rotations about three angles: roll, pitch and yaw. Thereby the rotation of {b} with respect to
the inertial frame {o} is represented by the Euler angles: roll (φ), pitch (θ), and yaw (ψ) follow-
ing aerodynamic conventions (54). In the inertial frame {o}, xo and yo describe the horizontal
plane and +zo points in the direction of gravity. On the other hand, the base frame {0} is a
rotated body frame {b} that allows the axis x0 to point laterally toward the right wing. This
rotated body frame is aimed at applying Denavit-Hartenberg convention (53) to place wing
frames.
51
4.2 Kinematics model
q3lh
lrlm/2
lwt
q4,5,6
q2
q1
z1
z2
z3
x4,5,6x3
x1
wingtip
y1
y3
zb
y2x2
xb
yb
zoxo
yo {o}
y4,5,6
z4,5,6z0
x0
y0
q3
q4,5,6q2
q7
xb
yb θ
φ
Figure 4.2: Detailed description of wing kinematics frames based on Denavit-Hartenberg (DH)
convention, qi corresponds to the rotation angle from axis xi−1 to xi measured about zi. The
subscript i indicates the frame of reference (i = 1..6). The inset is a top view of the right wing
showing planar angles.
Wing-frames
As mentioned, each wing, i.e., frames from {1} to {6} is treated as a serial chain of rigid bodies
connected to a base frame {0}. Frames {0} to {6} have been placed following modified Denavit-
Hartenberg (DH) convention (53). Denavit-Hartenberg parameters consists on four geometrical
parameters that specify the position of one coordinate frame relative to another. DH coordinate
frames are placed according to the following rules:
1. Axes z0 and zn+1 are aligned with the zb axes of the base.
2. Axes z1 to zn are aligned with the n joint axes such that zi is aligned with the axis of
joint i.
3. Axis xi is the common perpendicular between zi and zi+1, directed from zi to zi+1.
Having placed the coordinate frames, their relative locations are described by the following
DH parameters. Table 4.1 shows the numerical values of DH parameters whereas Figure 4.2
depicts how these geometrical parameters allows for the kinematics representation of the wing
structure. In the following the subscript i represents the joint frame of wing.
• αi, is the angle from zi−1 to zi measured about xi−1. For the application at hand, it
represents the body/bone twist.
• ai, is the distance from zi−1 to zi measured along xi−1. It represents the body/bone
length.
52
4.2 Kinematics model
Table 4.1: Modified Denavit-Hartenberg parameters per wing.
Joint Frame α a q d Constraint1
Shoulder2 1 π/2 +AoA lm/2 = 0.035m q1 0 120o
Shoulder 2 −π/2 0 q2 0 80o
Elbow 3 0 lh = 0.055m q3 0 60o
Wrist+digits 4,5,6 0 lr = 0.070m q4,5,6 0 45, 30, 18o
1 Absolute rotation range.2 AoA = angle of attack.
• qi, is the angle from xi−1 to xi measured about zi. It represents the body/bone rotation.
• di, is the distance from xi−1 to xi measured along zi. It represents the body/bone offset.
In Figure 4.2, axes z1 to z6 are aligned with the six joint axes such that zi is aligned with
the axis of joint frame {i}. Axis xi is the common perpendicular between zi and zi+1, directed
from zi to zi+1. The joint angles qi are defined from axis xi−1 to xi measured about zi. The
shoulder joint of the robot is composed by two degrees of freedom: q1, q2. The former angle
allows for the primary flapping motion (rotation about axis z1) whereas the latter allows the
wings to rotate about the axis z2 (cf. insert from Figure 4.2). The elbow joint has one degree
of freedom: q3, which allows the wings to contract or extend in sync with the flapping motion.
The wrist joint has three degrees of freedom: q4, q5, q6. Each angle allows for the rotation of
digits MCP-III, IV and V about axes z4, z5, z6 respectively. Both elbow and wrist joints provide
the morphing-wing capability to the robot.
Forward kinematics
Kinematics transformations that relate two consecutive frames of the wing are given by 4x4
homogeneous transformation matrix (Ti+1,i).
Ti+1,i =
[ri+1,i pi,i+1
0 1
]=
⎡⎢⎢⎣
cos qi − cosαi sin qi sinαi sin qi ai cos qisin qi cosαi cos qi − sinαi cos qi ai sin qi0 sinαi cosαi di0 0 0 1
⎤⎥⎥⎦ (4.1)
In Eq.(4.1) ri+1,i ∈ �3x3 is the basic rotation matrix that relates frames {i+1} onto frame
{i} and pi,i+1 ∈ �3 is the position vector that joints the frame i with i + 1. The terms αi,
53
4.2 Kinematics model
ai, di, qi, correspond to the geometrical DH parameters of the body i. Table 4.1 details these
parameters. The forward kinematics problem can be solved as 1:
Tn,0 =n∏i=0
Ti+1,i (4.2)
In this thesis the inertial model of the robot (cf. section 4.4) will be formulated by expressing
dynamics terms using six-dimensional (6D) vectors, which leads to the development of efficient
and portable code of the algorithms (55). For this reason kinematics quantities also must
be formulated using 6D operators. This approach is defined as spatial kinematics. Spatial
kinematics formulation is convenient for the application at hand since BaTboT has a floating
body that moves in �6 and the goal is to express how both angular and linear components of
physical quantities affect into the generation of spatial accelerations and forces of the body. To
this purpose spatial kinematics using 6D operators for rotation Ri+1,i and translation Pi,i+1
are defined as:
Ri+1,i =
[[ri+1,i]3x3 0
0 [ri+1,i]3x3
], Pi,i+1 =
[U [pi,i+1]3x30 U
], (4.3)
where U ∈ �3x3 is the identity operator, ri+1,i is the 3x3 rotation matrix, and pi,i+1 cor-
responds to the skew symmetric matrix corresponding to the vector cross product operator of
position vector pi,i+1. This matrix is described as:
pi,i+1 =
⎡⎣ 0 −pz py
pz 0 −px−py px 0
⎤⎦ (4.4)
Annex 10.1 shows the Matlab-code for the computation of spatial kinematics in �6. The
input of the algorithm corresponds to the wing joint trajectory, whose profile is defined by the
terms: joint positions (q), velocities (qd = q), and accelerations (qdd = q). It returns the spatial
positions (Xb), velocites (Vb) and accelerations (Vb) of the floating body. These spatial terms
are expressed with respect to the body frame {b}.
4.2.3 Wing trajectories and manuevers
Wing joint trajectory profile qi, qi, qi that allow forward and turning flight are extracted from
biological experiments in (20), (22), (23), (24). Figure 4.3a-b describe Cartesian paths for
both turning and forward flight, whereas plots c-d detail wing joint modulation scheme for the
1Detailed information related to rotation matrix properties, DH parameters, homogeneous transformations,
and spatial algebra applied to robotics, can be found in (55) and (91).
54
4.2 Kinematics model
q3
shoulder
q2
q4
q5 q6
q3
q4
q5
q6
elbow
wrist
MCP-III
IVV
(c) (d)
q1
xo
yo zo
yo
flight path
flight path
(a) (b)3 3
banked turn crabbed turn
{o}
{o}
(rolling) (heading)
2
2
q1=q2~=0
xo
yo{o}
xo
yo{o} zo
xo{o}
Figure 4.3: a)-b) Example of Cartesian paths during turning and forward flight, c-d) exam-
ple of wing modulation scheme for wing contraction during upstroke and wing extension during
downstroke. Source: the author.
corresponding maneuvering. It exemplifies how wing joints rotate for contracting and extending
the wing during flight. In the following, wing joint modulation patterns are introduced for both
forward and turning flight maneuvers.
Forward flight
The wingbeat cycle of a bat is composed by two phases: downstroke and upstroke. During
forward flight, both wings should flap symmetrically at the desired wingbeat frequency f . In
(22) a computational model for estimating the mechanics of horizontal flapping flight in bats is
presented. It quantifies (using high speed cameras) how markers placed in the bat wings move
in the three-dimensional space. It also calculates how most relevant joints of the wing rotate
during the wingbeat cycle. Figure 4.4 shows stills of the specimen Cynopterus brachyotis flying
at Brown wind tunnel facility1. Plot-a depicts the beginning of the wing downstroke. The body
1In-vivo experimental data and images of bat flight is copyright of Brown University. Wind-tunnel facility
is at Breuer Lab, School of Engineering http://brown.edu/Research/Breuer-Lab/index.html
55
4.2 Kinematics model
©Brown University
(a) (b)
(c) (d)
q4,5,6
markers
q2 q3
q1
Figure 4.4: Stills of wing kinematics during forward flight: a) beginning of downstroke, b) middle
downstroke, c) end of downstroke/beginning upstroke, d) middle upstroke. Source: (20), (22),
(23), (24).
of the specimen is lined up in a straight line, elbow joint is ∼ 89o. Plot-b depicts the middle
of the wing downstroke, where wings are extended to maximize the area aimed at increasing
lift force, at this state elbow joint is ∼ 60o. Plot-c depicts the end of the wing downstroke in
where the membrane is at maximum camber and the wings remain extended. Elbow joint is
∼ 30o. Plot-d depicts the middle wing upstroke where wings are folded to reduce drag. At this
state elbow joint is ∼ 85o. To know how the angle of the elbow joint rotates (q3) is essential
for the proper modulation of the wing shape. In bats the range of motion of the elbow accounts
for ∼ 90o but most of the rotations measured in (22) have shown an average range of motion
of ∼ 60o.
Kinematics of forward flight is shown in Figure 4.5. It only describes trajectory profiles
of the right wing since the left wing also moves with the same patterns. In (22), recordings
for horizontal flight have been carried out using three cameras placed in the wing-tunnel, as
represented by the cartoon of Figure 4.5 (top). The time change in the X, Y and Z positions
of the wrist joint and third MCP digit to wingtip (with respect to the body frame) were curve-
fitted by polynomials obtained using a least-squares algorithm (KaleidaGraph, version 3.0.5,
Abelbeck Software). In (22) eighth-order polynomials were used to increase precision in the
interpolation of MPC-III joints. Here, the MCP-III digit is considered as a rigid bone connected
to the wrist joint, thus third-order polynomial curves give enough precision for interpolating
56
4.2 Kinematics model
θ[r
ad]
θ[rad/s]
t[s]
θ[rd/s
2]
t[s]
X [m]
Y[m
]
x
y
z
(a) (c)
(b) (d)
downstrokeupstroke
q1 q2 q3 q4 q5 q6
q1 q2 q3 q4 = q5 = q6
q4 = q5 = q6q3q2q1
{1,2}{3} {4,5,6}
wristwingtip
IIIx b[m]
yb[m]
right-wing
cam2
cam1
cam3
q i,Rrad
[]
q i,Rrad
/s[
]q i,R
rad/s2
⎡ ⎣⎤ ⎦
1 2
3
©Brown University
Figure 4.5: (Simulation) Wing kinematics of forward flight (both wings move symmetrically at
wingbeat frequency of f = 2.5Hz): a) Cartesian trajectories of the wrist joint and the wingtip
frame. b) Joint angles. c)-d) Joint velocities and accelerations. For velocities, q3 = q4 (red plot)
and q5 = q6 (purple plot). For accelerations, q3 = q4, q5 = q6. Source: the author.
the Cartesian motion of both wrist and wingtip frames (axis Z does not account since only the
planar motion of both joints is considered). Polynomial values are shown in Table 4.2 whereas
Figure 4.5a shows the Cartesian path of wrist and wingtip frames that are presented by the
polynomials.
To achieve forward flight the wing joints are modulated as described in Figure 4.5b. It shows
the trajectory profile of each wing joint (qi) during a wingbeat cycle (f = 2.5Hz). In the model,
the primary flapping motion (shoulder joint q1) is simply generated as q1 = 60cos(2πf) allowing
the wings to flap an angle range of of 120o at the desired wingbeat frequency f . Similarly the
joint profile q2 = −10cos(2πf−10) allows the wings to rotate forward and backward around the
57
4.2 Kinematics model
Table 4.2: (Forward flight) coefficients of the third-order polynomial curves that describe Carte-
sian paths of wrist and wingtip frames fitted to the xb, yb coordinates of the body frame.
Frame X Y
0.0550 + 0.0475t− 0.0075t3 0.0950− 0.0389t0.0039t3
0.0950 + 0.0250t− 0.0224t2 + 0.0074t3 0.0600− 0.0271t+ 0.0118t2 − 0.0097t3
Wrist 0.1000 + 0.0004t− 0.0240t2 − 0.0118t3 0.0600 + 0.0323t+ 0.0138t2 + 0.0155t3
0.0750− 0.0122t+ 0.0098t2 + 0.0113t3 0.0800 + 0.0132t− 0.0072t2 − 0.0070t3
0.0550− 0.0265t− 0.0033t3 0.0950 + 0.0198t+ 0.0024t3
0.1800 + 0.0406t− 0.0056t3 0.0900 + 0.0124t− 0.0124t3
0.2150 + 0.0237t− 0.0169t2 + 0.0082t3 0.0900− 0.0249t− 0.0373t2 + 0.0122t3
Wingtip 0.1600− 0.0470t+ 0.0129t2 − 0.0121t3 −0.0350− 0.0077t+ 0.0114t2 − 0.0082t3
0.1750 + 0.0150t− 0.0257t2 − 0.0129t3 0.0100 + 0.0396t− 0.0066t2 − 0.0060t3
0.1600 + 0.0022t− 0.0257t2 + 0.0086t3 0.0500 + 0.0444t+ 0.0022t3
gravity axis within an angle range of 20o. On the other hand, the change of wing morphology
is driven by the joint profiles q3...q6. The elbow joint (q3) is key for contracting and extending
the wing during flight. In (22) it was shown that the wingbeat cycle can be partitioned into
approximately 65% of downstroke time required for wing extension and 35% of upstroke time
required for wing contraction, i.e., the wings contract faster to reduce the generation of drag
forces. Here, this approach is achieved by modulating the elbow joint as detailed in Figure 4.5b
(red plot): wings extend in about 0.25s of downstroke time (62.5%) and contract in about 0.15s
of upstroke time (37.5%). Also note how digits III, IV and V (driven by joints q4, q5, q6) open
and close maintaining the minimum angle proportions previously determined in Figure 3.7b:
(downstroke) q4 = 15o, q5 = 65o, and q6 = 128o, (upstroke) q4 = 90o, q5 = 95o, and q6 = 110o.
These proportions contribute to the proper tension of the wing membrane during the process
of extension and contraction.
Turning flight
Kinematics of turning flight is shown in Figure 4.3. In (92), kinematics data of slow turn
maneuvering was recorded as represented by the cartoon of Figure 4.3 (top). To successfully
complete a turn, bats must translate its center of mass along the flight path (i.e. change its
flight direction) and rotate its body around its center of mass to align its body orientation with
the new direction. The magnitude of change in direction of flight is a function of the impulse
(force-time) perpendicular to the original direction of movement. Impulse is the result of the
centripetal force produced by the change of the orientation of the net forces generated by the
58
4.2 Kinematics model
q i,R
(rad)
t(s)
q i,R
(rad/s)
t(s)
¨q i
,R(r
ad/s
2)
t(s)
q i,L
(rad)
t(s)
q i,L
(rad/s)
t(s)
Yb(m)
xb(m
)
¨q i
,L(r
ad/s
2)
t(s)
x
y
zy
z
x
left-wing right-wing
wristwingtip
{0,b}
{1,2}{1,2}{3} {3}
{4,5,6}{4,5,6}
q1 q2 q3 q4 q5 q6
q1 q2 q3 q4 q5 q6
q1 q2 q3 q4 = q5 = q6
q1 q2 q3 q4 = q5 = q6
q4 = q5 = q6q3q2q1
q4 = q5 = q6q3q2q1
(a)
(b) (c) (d)
(e) (f) (g)
yo
xo
zoTop view
Flight path
cam1
cam2
cam3
downstrokeupstroke
©Brown University
markers tracked by cameras allow kinematics quantification
Figure 4.6: (Simulation) Wing kinematics of turning flight at wingbeat frequency of f = 2.5Hz):
a) Cartesian trajectories for the wrist joint and the wingtip frame of each wing (top view). b)-c)-d)
Left wing more contracted: joint angles, velocities and accelerations. e)-f)-g) Right wing more
extended: joint angles, velocities and accelerations. For velocities, q3 = q4 (red plot) and q5 = q6
(purple plot). For accelerations, q3 = q4, q5 = q6. Source: the author.
body and wings. Net forces are the sum of aerodynamic forces (lift and drag) and also inertial
forces (thrust and weight). To turn, bats bank (roll) around its cranio-caudal axis, tilting the
vector of the vertical component of the net aerodynamic force (i.e. lift in level flight) laterally
and towards the center of the turn. The reorientation of net forces produces a laterally oriented
force that drives the bats into a turn.
Based on this foundation, turning flight can be achieved by generating roll momentum by
means of wing contraction and extension, which allows for the displacement the center of mass
of the robot towards the expanded wing. This approach allows the robot to roll (φ). The
59
4.2 Kinematics model
q3
end of wing extension
t =0.4
tR,down = 0.25 tR,up = 0.15
tR,down − tR,up
q4
t =0 t =0.25t =0.15
q3,L
q3,R
q4,L
q4,R
Figure 4.7: Comparison between wing joint trajectories of left and right wings (q3 and q4)
considering a morphing-wing factor of fmc = 0.1. Source: the author.
modulation of the wing joints follows the same trajectory patterns than those presented in
forward flight, however, to achieve the rolling motion one wing should contract and extend is
less proportion compared to the other. This proportion is called the morphing-wing factor fmc.
The influence of the morphing-wing factor can be appreciated in the Cartesian paths of the
wrist and wingtip of the left wing (cf. Figure 4.6a). Cartesian paths differ from the right wing
because the wing joints of the left wing q3,L..q6,L are being modulated with different morphing-
wing factor than the wing joints of the right wing q3,R..q6,R (cf. Figures 4.6b-e). Term fmc is a
number between > 0 and <= 1. If fmc = 1 both wings contract and extend symmetrically in
the same proportion as forward flight does. In Figure 4.6b the left wing has a morphing-wing
factor of fmc = 0.1. Figure 4.7 details the joint motion of q3 and q4 by comparing both left and
right wing profiles from Figures 4.6b-e.
Term fmc puts out of phase and reduces the amplitude of profile q of one wing with respect
to the other. This causes the wings have different contraction and extension periods during
downstroke and upstroke. Note in the case of Figure 4.7, the right wing (q3,R and q4,R) extends
in tR,down = 0.25s and contracts in tR,up = 0.15s but in order to generate inertial forces that
cause de body to roll, the left wing cannot be modulated in the same manner. Thus, the wing
factor is applied to change the modulation of the left wing aimed at increasing the rolling torque
towards the right wing. This is done by q3,L = fmcq3,R (same procedure for q4). This approach
not only reduces the amplitude of q3,L (keeping the left wing more contracted) but also reduces
the extension period of q3,L due to fmc = 0.1 = tR,down − tR,up, making the left wing to
60
4.3 Inertial model
{i}=1
{cm}
si,cm
pi,i+1 {i+1}
body i
body i+1
zi
xiyi
xi+1
yi+1
......i=n=6
......
base
inertial frame
{o}
zo
xoyo
qi
qi+1
{0}
x0y0
z0
zi+1
x
yz
body i
vi{i}
ωifi i
{cm}
......
......
zb
xb
ybbody frame {b}
τφ
τθ
right wing
left wing
i=n=6
{i}=1
{i+1}
zipi,i+1
xi
yi
yi+1
xi+1
zi+1
qi
Fi,Li=n
0
∑ Fi,Ri=n
0
∑
qi+1
τ i
Figure 4.8: Rigid multibody serial chain that composes each wing. Spatial forces of each body
contain both linear fi and angular τi force components stacked into a six-dimensional vector Fi.
These forces are propagated from the wingtip {i} = n to the base frame {0}. Subscripts R,L
denote for right and left wing respectively. The resultant spatial force (FT ) acting on the base
frame {0} is the sum of spatial forces generated by both wings. The inset shows the velocity of a
rigid body i expressed in terms of ωi and vi, and the force acting on a rigid body i expressed in
terms of fi and τi.
decrease aerodynamic forces and therefore to increment the effects of wing inertia modulation
into the production of rolling torque. The rolling torque is mainly caused by the inertial forces
produced by each wing at the center of mass of the body. To analyze the inertial effects of wing
modulation for achieving forward and turning flight, the following section introduces the inertial
model of BaTboT. The inertial model calculates the net inertial forces that are generated at
the center of mass of the robot due to the propagation of forces produced by the wings when
moving using the kinematics profile presented in this section. It also calculates the spatial
accelerations of the body based on applied forces.
4.3 Inertial model
4.3.1 Spatial notation
Equations of Motion (EoM) are formulated based on Newton-Euler formalism using spatial
operator (55). Spatial operators leads to six-dimensional physical quantities that combine the
angular and linear aspects of rigid-body motions and forces. Using spatial algebra notation,
61
4.3 Inertial model
Table 4.3: Spatial operators.
Description 6D operator
Spatial velocities of body i with respect to joint frame {i} Vi = [ωxωyωzυxυyυz ]T ∈ �6x1
Spatial accelerations of body i with respect to joint frame {i} Vi = [ωxωyωz υxυy υz ]T ∈ �6x1
Spatial forces of body i with respect to joint frame {i} Fi = [τxτyτzfxfyfz ]T ∈ �6x1
Spatial rotation from joint frame {i+ 1} to {i} Ri+1,i =
[ri+1,i 0
0 ri+1,i
]∈ �6x6
Spatial translation from joint frame {i} to {i+ 1} Pi,i+1 =
[U pi,i+1
0 U
]∈ �6x6
Spatial translation from joint frame {i} to center of mass {cm} Si,cm =
[U si,cm
0 U
]∈ �6x6
Projection onto the axis of motion of joint i H =[
0 0 1 0 0 0]T ∈ �6x1
Spatial inertia of body i with respect to {cm} Ii,cm =
[Ji,cm 0
0 miU
]∈ �6x6
Inertial tensor of body i with respect to {cm} Ji,cm =
⎡⎢⎣ Ixx 0 0
0 Iyy 0
0 0 Izz
⎤⎥⎦ ∈ �3x3
velocities Vi, accelerations Vi, and forces Fi of a rigid body i are expressed with respect to its
joint frame {i} using six-dimensional vectors, as:
Vi =
[ωiυi
], Vi =
[ωiυi
], Fi =
[τifi
]∈ �6x1 (4.5)
The inset in Figure 4.8 details how the physical components in Eq. (4.5) are expressed in
a rigid body i. In order to propagate the spatial forces produced by the right (Fi,R) and left
(Fi,L) wings onto the base frame {0}, EoM can be recursively solved as:
FT =1∑i=6
Fi,R +1∑i=6
Fi,L ∈ �6x1 (4.6)
FT in Eq. (4.6) is the sum of propagated forces of both wings. The subscripts R and L
refer right and left wing respectively. To propagate Fi,R and Fi,L spatial operators for rotation
Ri+1,i ∈ �6x6 and translation Pi,i+1 ∈ �6x6 in Eq. (4.3) are used. Table 4.3 summarizes all
spatial operators used for the dynamics description of the inertial model. Similarly to Pi,i+1
the operator Si,cm ∈ �6x6 represents a spatial transformation for translation from joint frame
{i} onto the frame located at the center of mass of the body {cm}. The skew symmetric matrix
(si,cm) corresponds to the vector cross product operator of position vector si,cm. Also, each
rigid body with scalar massmi and inertial tensor Ji,cm ∈ �3x3 is composed by a spatial inertial
operator Ii,cm ∈ �6x6 with respect to center of mass (CM) of the body i. The tensor Ji,cm is
composed by moments of inertia (Ixx, Iyy, Izz) which are calculated taking the volume integral
62
4.3 Inertial model
by the square perpendicular distance from the corresponding axis, cf. Eq. (4.7). The term ρ is
the density of the rigid body i. Moments of inertia physically represent the resistance that the
body presents when is accelerating.
Ixx =∫ ∫ ∫ (
y2 + z2)ρdv,
Iyy =∫ ∫ ∫ (
x2 + z2)ρdv,
Izz =∫ ∫ ∫ (
x2 + y2)ρdv,
(4.7)
4.3.2 Equations of motion (EoM)
To solve FT in Eq. (4.6) the term (1∑i=n
Fi) must be derived for each wing, being the subscript
i an indicator of the rigid body: the shoulder, humerus, radius, MCP-III, etc. Spatial forces
are calculated with respect to the joint frame {i} of the rigid body i and then propagated and
oriented towards the next body in the serial chain (cf. Figure 4.8). Spatial operators from
Table 4.3 are used to express EoM, which basically consist on the calculation of forces that
must be applied to the wing-system (Fi) in order to produce a given joint acceleration response
(qi). This is known as the inverse dynamics problem. Therefore, spatial forces and velocities
calculated with respect to the center of mass {cm} of a rigid body i are related with respect to
the joint frame {i} of the body, as:
Fi = STi,cmFi,cm,Vi = STi,cmVi,cm
(4.8)
By definition the spatial force acting on the center of mass of a body i can be expressed by
differentiating the inertial moment (Li,cm = Ii,cmVi,cm) with respect to time, as:
Fi,cm = Li,cm = Ii,cmVi,cm + Ii,cmVi,cm (4.9)
Replacing Eq. (4.9) into Fi in Eq. (4.8):
Fi = STi,cm
[Ii,cmVi,cm + ξ
](4.10)
Where ξ = Ii,cmVi,cm is the gyroscopic force acting on the center of mass frame {cm} of the
rigid body i. The spatial acceleration Vi,cm can be solved by differentiating Vi,cm in Eq. (4.8)
with respect to time, as:
Vi,cm = Si,cmVi,
Vi,cm = Si,cmVi + Si,cmVi(4.11)
63
4.3 Inertial model
Replacing Vi,cm in Eq.(4.11) into Fi in Eq.(4.10) :
Fi = STi,cm
[Ii,cm(Si,cmVi + Si,cmVi) + ξ
]. (4.12)
The term Si,cmVi contains Coriolis effects. The spatial forces in Eq. (4.12) can be recur-
sively propagated along the wing by considering the spatial operators for rotation Ri+1,i and
translation Pi,i+1. Therefore, spatial forces in Eq. (4.12) are backward propagated, from the
wingtip (i = n = 6), to the base frame {0} (i = 1), as:
1∑i=6
Fi = Ri+1,iPTi,i+1Fi+1 + STi,cm
[Ii,cm(Si,cmVi + Si,cmVi) + ξ
]= Ri+1,iP
Ti,i+1Fi+1 + STi,cmIi,cmSi,cmVi + STi,cmIi,cmSi,cmVi + STi,cmIi,cmVi,cm
(4.13)
The expression Ri+1,iPTi,i+1Fi+1 allows for the projection of spatial forces along the serial
chain of bodies that compose the wing structure. Also note from Eq. (4.13) that the term
STi,cmIi,cmSi,cm refers to the spatial inertia Ii calculated with respect to the joint frame {i} by
applying the parallel axis theorem. Equation (4.13) is rewritten as:
1∑i=6
Fi = Ri+1,iPTi,i+1Fi+1 + IiVi +
[IiSi,cm + Ii
]Vi. (4.14)
To complete the solution of the inertial model in Eq. (4.14), the set of spatial velocities (Vi)
and accelerations (Vi) are also recursively calculated as:
Vi = Pi,i+1RTi+1,iVi−1 +Hqi
Vi = Pi,i+1RTi+1,iVi−1 + Pi,i+1R
Ti+1,iVi−1 +Hqi + Hqi
(4.15)
The term H allows for the projection of the joint trajectory profile qi,qi onto the axis of
motion. Wing joint trajectories are presented in Figure 4.6. Algorithm 1 describes how to
solve FT in Eq. (4.6) and Annex 10.2 details the Matlab-code for solving Algorithm 1. The
inputs of the algorithm are the wing joint profiles (qi, qi, qi) and the kinematics velocity (Vb) and
acceleration (Vb) of the body calculated from kinematics algorithm in Annex 10.1. Algorithm
1 returns the spatial forces FT and accelerations of the floating body Vb that are produced by
FT . Both terms are with respect to the body frame {b} due to the applied rotation R0,b. It
rotates Fi that is with respect to the frame {0} by R0,b = rz0(−π/2)rx0(π) (cf. Figure 4.2).
Vb =
(Ib +
[R0,b
0∑i=6
(Ri+1,iPTi,i+1Ii,R)+R0,b
0∑i=6
(Ri+1,iPTi,i+1Ii,L)
])−1
FT (4.16)
64
4.3 Inertial model
Algorithm 1 Inertial model computation
initialize:
k = 1, V0 ←[
φ θ ψ 0 0 0], V0 ←
[φ θ ψ 0 0 9.81
]while k ≤ 2, do –for each wing–:
——————————————————-
1. Forward recurrence: spatial velocities and accelerations
for i = 1 → n do
Vi ← Pi,i+1RTi+1,iVi−1 +Hqi
Vi ← Pi,i+1RTi+1,iVi−1 + Pi,i+1R
Ti+1,iVi−1 +Hqi + Hqi
end for
——————————————————-
if (aerodynamics are included) do Fi+1 ←[
0 0 0 −FD 0 FL
]Telse do Fi+1 ←
[0 0 0 0 0 0
]T2. Backward Recurrence: spatial forces
for i = n → 1 do
Fi ← Ri+1,iPTi,i+1Fi+1 + IiVi +
[IiSi,cm + Ii
]Vi //forces
τi ← HTFi //torques
if i! = n do Ii + [Ri+1,iPTi,i+1]Ii+1[Ri+1,iPi,i+1] //inertia propagation
end for
if k == 1 do Fi,R ← Fi, Ii,R ← Ii //(R): right wing
else do Fi,L ← Fi, Ii,L ← Ii //(L):left wing
k ++
end while-do
——————————————————-
3. Compute body forces with respect to body frame {b}FT ← R0,bFi,R +R0,bFi,L
4. Compute body inertia Ib with respect to body frame {b}Ib +
[R0,b
0∑i=6
(Ri+1,iPTi,i+1Ii,R)+R0,b
0∑i=6
(Ri+1,iPTi,i+1Ii,L)
]5. Compute body acceleration caused by FT with respect to body frame {b}Vb =
(Ib +
[R0,b
0∑i=6
(Ri+1,iPTi,i+1Ii,R)+R0,b
0∑i=6
(Ri+1,iPTi,i+1Ii,L)
])−1
FT
Return FT , Vb
4.3.3 Rolling and pitching torques
To achieve forward and turning flight, section 4.2.3 showed how to kinematically take advan-
tage of the morphing-wing modulation aimed at changing the wing shape. Dynamically, these
changes in wing shape induce effective forces (torques) at the center of mass of the robot that
allow the body to rotate. Here, the inertial model in Eq. (4.14) is used to define rolling (τφ)
and pitching torques (τθ) components that induce forward and turning flight respectively. Both
terms are extracted from the components of FT as:
τφ =[1 0 0 0 0 0
]FT
τθ =[0 1 0 0 0 0
]FT
(4.17)
Rolling and pitching torques from Eq. (4.17) are with respect the body frame {b} as shown
65
4.4 Aerodynamics model
FT(x)
FT(z)
FL
FD
Fnet
Fnet
Figure 4.9: Free-body diagram of a bat in accelerating flight, indicating the aerodynamic and
gravitational forces that accelerate the center of mass (COM). Lift is perpendicular to the direction
of flight whereas drag and thrust are parallel to the direction of flight. The net force produced
can be decomposed into net force components parallel and perpendicular to the direction of flight
(see inset). The parallel component corresponds to the net thrust. Thus, measurements of the
acceleration of the COM would directly reflect the net forces acting on it. Source: (24).
in Figure 4.8. Simulations in section 4.6.2 show estimations about the influence of wing inertia
into the production of rolling and pitching torques.
4.4 Aerodynamics model
Figure 4.9 shows free-body diagram of a bat in accelerating flight, indicating the aerodynamic
and inertial forces that the center of mass of the robot. Inertial forces i.e., thrust and weight
components, can be directly modeled using the inertial model (FT ) from Eq. (4.14) by ex-
tracting the force components of the Zb and Xb axes respectively. Aerodynamics forces i.e., lift
and drag components, can be directly quantified by measuring the force produced by the real
robotic platform. It is more accurate to have an identified model of aerodynamics forces rather
than a simulated model using fluid dynamics theory. Experiments to quantified aerodynamic
forces will be carried out in Section 7.3 of Chapter 7. For instance, this section provides the
basic foundations to measure aerodynamics and net forces.
4.4.1 Lift and drag forces
Lift is perpendicular to the direction of flight whereas drag and thrust are parallel to the
direction of flight. Figure 4.10a shows the testbed, which shows BaTboT (in the inset)1 mounted
1Due to aerodynamics modeling depends strictly on real experiments aimed at quantifying lift and drag
coefficients, this section gives a brief insight into the experimental tests to carry out using the final BaTboT
66
4.4 Aerodynamics model
(b)
CL
CD
(a)
AoA
drag (FD)thrust (FT,x)
FT,z
lift (FL)
{o}
zo
yoxoFnet
Force sensor
AoA [deg]
Figure 4.10: (Experimental) aerodynamics identification –wind-tunnel–: a) the bat-robot is
mounted on top of a 6-DoF force sensor from which both lift FL and drag FD forces are experi-
mentally calculated as a function of the airflow speed and angle of attack (AoA), b) Lift and drag
coefficients (CL, CD) calculated from measured lift and drag forces. Source: the author.
in the wind tunnel, on the end of a supporting sting that defines the angle of attack (AoA).
The robot is mounted on top of a 6-DoF force sensor1 from which both lift CL and drag CD
coefficients are calculated from the force measurements of lift and drag forces.Typical results
are shown in Figure 4.10b. Both lift (L) and drag (D) force components are measured directly
from the force sensor by applying Eq. (4.18).
L = FLcos(AoA)− FDsin(AoA)D = FDcos(AoA) + FLsin(AoA)
(4.18)
Then, the lift and drag coefficients are computed as:
CL = 2L(ρV 2airAb)
−1 CD = 2D(ρV 2airAb)
−1 (4.19)
Where the term ρ = 1.20Kgm3 is the air density, Vair is the airspeed, and Ab = 0.05m2 is
the planform area of the wing. Using Eq.(4.18) and (4.19) identified aerodynamic forces can be
incorporated into the inertial model of the robot, as detailed in Algorithm 1 (cf. before step 2).
At the beginning of the force propagation (i = 6), the term Fi+1 = 0 in the absence of external
forces, i.e., accounting only inertial contribution. In case aerodynamics forces are included into
the inertial model, the term Fi+1 would correspond to:[0 0 0 −FD 0 FL
].
prototype. Further experiments with detailed analysis of aerodynamics behavior is presented in Chapter 7
section 7.3.1force sensor Nano17 transducer ATI Industrial Automation, 0.318 gram-force of resolution, http://www.
ati-ia.com/
67
4.4 Aerodynamics model
4.4.2 Net forces
As explained by (24), to accelerate during forward flight, any flying organism must produce a net
aerodynamic force to counteract gravity and overcome drag. This force can be decomposed into
a net force component in the direction of flight that corresponds to the difference between thrust
and drag, i.e. net thrust, and a perpendicular component that corresponds to the difference
between lift and weight (cf. Figure 4.9). During forward, steady flight, the average lift over the
course of a wingbeat must equal body weight, and average thrust must equal drag. However,
unlike airplanes, flying organisms cannot continuously generate constant lift and thrust because
of the oscillatory nature of flapping, so instantaneous force generation varies across the wingbeat
cycle. As a consequence, a flying bat will accelerate and decelerate throughout a wingbeat, even
during steady-state flights where average acceleration is zero over the complete cycle. In (24) is
demonstrated that in slow flight bats generate a net forward force during the upstroke: During
upstroke, the upward and backward acceleration of the wings will produce an inertial force
that will move the body forward and downward with respect to the downstroke. This force
will produce a forward-oriented component, or inertial thrust (FT,x) during upstroke. During
downstroke, the downward and forward acceleration of the wings will produce an inertial force
that will move the body backward and upward while keeping the position of the center of mass
constant. The horizontal component of this inertial force will produce negative inertial thrust
during downstroke (−FT,x). So bats produce positive inertial thrust during the upstroke motion
of the wings mainly due to the body moves in opposition to the flapping direction in order to
conserve momentum. Bats properly modulate wing kinematics to take a maximum advantage
of wing inertia on the production of body accelerations. These accelerations are produced by
the net forces (Fnet), which are calculated with respect to the body frame {b}, as:
Fnet =([
0 0 0 1 0 0]FT − FD
)+(FL − [
0 0 0 0 0 1]FT
)(4.20)
Inertial forces are likely to be significant in bats because the mass of the wings comprises
a significant portion of total body mass, ranging from 11% to 33% and because wings undergo
large accelerations. In this specie, wings’ mass accounts for 31% of the total mass, while in the
robot wings’ mass accounts for 37% of the total mass. In this thesis it will be demonstrated
the significant impact of wing modulation kinematics into the production of net forces. By
properly controlling wing modulation, both inertial (FT ) and aerodynamics forces (FL, FD) will
be efficiently produce aimed at generating large net forces (Fnet) during forward flight.
68
4.5 SMA muscle-like wing actuation
SMA
spring
SMA1
SMA2
rj rj
lsma
SMA1
SMA2-+
I1 Fsma-1
Fsma-2
I2τ 3
τ 3τ 3
q3
SMA1
-+
I1 Fsma-1τ 3
lsp −k ∗ lsp(a) (b)
Biceps
Triceps
Figure 4.11: SMA actuation configurations: a) (top) SMA joint with bias spring concept,
(medium) mechanic implementation, (below) SMA-spring model representation. b) (top) SMA
antagonistic joint, (medium) mechanic implementation that mimic how bicep and tricep muscles
operate, (below) antagonistic pair of SMAs model representation. In both configurations SMA
wires extend along the humerus bone of BaTboT wings, acting as artificial muscles that pull the
elbow joint q3. SMA pulling forces (Fsma) produce a joint torque (τ3) that rotates the elbow,
pulling in the ”fingers” to slim the wing profile on the upstroke. Source: the author.
4.5 SMA muscle-like wing actuation
Recently, Shape Memory Alloys (SMAs) have opened new alternatives and the potential of
building lighter and smaller smart actuation systems (43), (44), (45), (93), (46), (94). To closely
mimic the morphing-wing apparatus of bats, extremely light wires with negligible volume acting
as artificial muscle fibers seem to be an adequate solution. To this purpose SMAs wires able to
contract upon electrical heating have been used as artificial biceps and triceps that allow the
wing to contract and extend by means of elbow and wrist joints rotation. This section provides a
comprehensive review of SMA modeling, highlighting the thermo-mechanical properties behind
NiTi material aimed at defining equations that partially govern phenomenological behavior, i.e,
shape memory effect, hysteresis, strain, stress, etc. To this purpose a phenomenological model
is presented in order to assess the performance and limits of SMA technology for the application
at hand.
69
4.5 SMA muscle-like wing actuation
4.5.1 SMA actuation configurations
Upon heating SMA wires use the one-way shape memory effect to generate force and motion.
Because SMA actuators can only contract in one direction, it is necessary to provide a biasing
force to return to the neutral position. This can be accomplished using a bias spring (cf. 4.11a),
or another SMA element in an antagonistic arrangement (cf. 4.11b). In practice the former is
mostly used, but the latter gives further control range.
In the SMA actuators with bias spring arrangement, only one SMA is heated and cooled, so
the hysteresis effect has quite a significant influence on control performance. This configuration
has the advantage of having no slack issues over the SMA or stress when recovering to the neutral
position. On the contrary the antagonistic configuration, which heats one actuator while the
other cools, can reduce the hysteresis effect, as experimentally demonstrated by (73), and (82).
The advantage of this configuration can be appreciated from a control perspective since instead
of providing passive biasing force or motion, both directions can be actively controlled. This
increases the range of controllable actuation at expense of adding a phenomenon called the two-
way shape memory effect which is mainly produced when the wires extend upon cooling. The
passive SMA wire can develop a few millimeters of slack as it cools, which consequently affects
the accuracy of the control. The two-way shape memory effect becomes even more problematic
in the antagonistic arrangement of SMA actuators, and may lead to slower response and even
wire entanglement. Since this is a control problem, section 6.5 in Chapter 6 will approach these
issues.
For instance, the following subsection focuses on describing how to properly model thermo-
mechanical properties of SMAs and its use as muscle-like actuators. This allows to assess and
explore the limits to increase SMA performance in terms of actuation speed and output torque
by keeping the input power below the limits of overheating.
4.5.2 SMA phenomenological model
SMAs exhibit an unique thermomechanical property due to the phase transformation of the
material, from austenite phase to martensite phase and vice versa. These transformations
mainly occur due to changes in temperature and stress. Extensive research has been devoted
to model these properties. Tanaka in (71) was one of the pioneers to study a stress-induced
martensite phase transformation, proposing an unified one-dimensional phenomenological model
that make use of three state variables to describe that process: temperature T , strain ε, and
70
4.5 SMA muscle-like wing actuation
martensite fraction ξ. His main contribution was to demonstrate that the rate of stress is a
function of strain, temperature and martensite fraction rates. Later, Brinson (78) improved
on Tanaka’s model by separating the calculation of the martensite fraction into two parts,
one induced by stress and the other one induced by temperature. This issue allowed for the
description of the shape memory effect at low temperatures.
Elahinia (56),(70) proposed an enhanced phenomenological model compared to the previous
ones, and also addressed the nonlinear control problem. This model was able to better describe
the behavior of SMAs in cases where the temperature and stress states changed simultaneously.
Their model was verified against experimental data regarding a SMA-actuated robotic arm. As
a result, the phenomenological model was able to predict SMA behavior also under complex
thermomechanical loadings. Further experiments were also carried out in (95).
In this thesis, Elahinia’s phenomenological model (70) has been used for assessing the limits
of SMA operation. The model consists of four parts: i) heat transfer, ii) mechanics model,
iii) forward/reverse phase transformation, and iv) kinematics model. As shown in Figure 4.11,
the input of the model is the electrical current Isma to drive the SMA and the output is the
elbow joint rotation q3 that is produced by the strain rate of the SMA when pulling the joint.
This model allows for determining proper parameters to safe overload SMA performance without
compromising physical damage to the shape memory effect or overheating issues when subjected
to high amount of input power. Simulations in section 4.6.4 are carried out for characterizing
overload SMA actuation response that would be fundamental for control tuning and experiments
with the real platform.
Heat transfer model
The SMA wire heat transfer equation consists of electrical (Joule) heating and natural convec-
tion:
msmacpT = I2smaRsma − hcAc (T − To) (4.21)
SMA NiTi wires have a diameter of 150μm, a mass per unit length of msma = ρπr2j where ρ
is the density of wire, 2rj is diameter of wire, Ac = π2rj is circumferential area of the unit length
of the wire, cp is specific heat, Isma is applied electrical current, Rsma is electrical resistance
per unit length of the wire, T is temperature of the wire, To is the ambient temperature, and
hc is the heat convection coefficient. Although in Eq. (4.21) is assumed that hc and Rsma are
both constant, a detailed experimental analysis on the resistance variation of the SMAs can be
71
4.5 SMA muscle-like wing actuation
found in Section 6.4. Using Eq. 4.21 is possible to model how the NiTi wire would heat upon
electrical current and by removing the term I2smaRsma (heating power), the equation can be
also used to model how the NiTi wire cools in the absence of heating power.
Mechanical model
SMA mechanical model was firstly introduced by Tanaka in (71). It relates stress rate (σ) with
temperature rate (T ) as:
σ =θs−Ω(Af−As)
−1
1−Ω(Af−As)−1Cm
T (4.22)
Where θs corresponds to the thermal expansion factor of the wire, Ω is the phase transfor-
mation factor, Af , As are the austenite final and initial temperatures and Cm is the effect of
stress coefficient on martensite temperature. Also, the strain rate (ε) during heating phase can
be calculated as:
ε = σ−θsT−ΩξEA
(4.23)
Where EA is the austenite the Young’s modulus and ξ is the phase transformation rate
which is presented in the following.
Phase transformation model
The reverse transformation equation that describes the phase transformation from martensite
to austenite during heating is:
ξ = ξm2 [cos (aA (T −As) + bAσ) + 1] (4.24)
where ξ is martensite fraction that has a value between 1 (martensite phase) and 0 (austenite
phase). The terms aA = π(Af − As)−1 and bA = −aAC−1
A are the curve-fitting parameters
of the phase transformation. Also, the forward transformation equation describing the phase
transformation from austenite to martensite during cooling is:
ξ = 1−ξa2
[cos (aM (T −Mf ) + bMσ) +
1+ξa2
](4.25)
Where aM = π(Mf −Ms)−1 and bM = −aMC−1
M are the curve-fitting parameters, where
Mf ,Ms are the martensite phase final and initial temperature respectively.
72
4.5 SMA muscle-like wing actuation
Table 4.4: Parameters for SMA phenomenological model
Variable Model Parameters Value [unit]
Temperature
Heating: msma,Rsma, Isma 1.14 × 10−4 [Kg], 8.5 [Ω]
msmacpT = I2smaR − hcAc (T − To) Ac 1.76 × 10−8[m2
]
Cooling: hc 150[Jm−2◦C−1s−1
]
(T ) msmacpT = −hcAc (T − To) Cp 0.2[KcalKg−1◦C−1
]
Stress (σ)
Heating: Ω −1.12 [GPa]
σ =θs−Ω
(Af−As
)−1
1−Ω(Af−As
)−1Cm
T θs 0.55[MP◦aC−1
]
Cooling: Cm,Ca 10.3[MP◦aC−1
]
σ =θs−Ω
(Ms−Mf
)−1
1−Ω(Ms−Mf
)−1Ca
T As,Af ,Ms,Mf 68, 78, 52, 42[◦C]
Strain (ε)
Heating:
ε =σ−θsT−Ωξ
EAEA 75 [GPa]
Cooling: EM 28 [GPa]
ε =σ−θsT−Ωξ
EM
FM (ξ)
Heating: ξm, ξa 1, 0 [dimensionless]
ξ =ξm2
[cos
(aA (T − As) + bAσ
)+ 1
]aA 0.31
[◦C−1]
Cooling: aM 0.31[◦C−1
]
ξ =1−ξa
2
[cos
(aM
(T −Mf
)+ bMσ
)+
1+ξa2
]bA, bM −0.03
[◦C−1]
Kinematics model
The kinematic model describes the relationship between SMA strain rate ε with the angular
rate of the elbow joint q3:
q3 = lsmaε(2rj)−1, (4.26)
The rotation of the elbow joint (q3) can be calculated by integrating Eq. (4.26) with respect
to time, being lsma the length of the SMA wires, and rj the radius of the joint (cf. Figure
4.11b).
SMA phenomenological algorithm
Table 4.4 summarizes the parameters used for the simulation of the thermo-mechanical equa-
tions. Further details on the values assigned to most coefficients can be also found in (56) and
(70).
Algorithm 2 describes the procedure for computing the thermo-mechanical equations of the
SMA phenomenological model. Also Annex 10.4 shows the Matlab-code for Algorithm 2. Inputs
are the electrical currents (Isma) to drive each SMA actuator. The output is the elbow joint
rotation q3. Step 1 computes the temperature rate of the SMA (T ) based on current inputs.
It allows for the evaluation of overheating when the SMAs are subject to high values of input
currents. In step 2 SMA stress rate is calculated (σ). Whether the temperature on SMA wires
increases dramatically, stress would also increase allowing for the identification of over-stressing
73
4.6 Simulation and experimental results
Algorithm 2 SMA thermo-mechanical computation
1. Compute SMA temperature rate: T ← m−1smac
−1p (I2smaRsma − hcAc (T − To))
2. Calculate the SMA stress rate upon heating: σ ← θs−Ω(Af−As)−1
1−Ω(Af−As)−1
CmT
3. Calculate phase transformation rate (differentiating Eq. (4.24) with respect to time):
ξ ← − ξm2
[sin (aA (T −As) + bAσ) + (aAT + bAσ)
]4. Compute SMA strain rate upon heating: ε ← σ−θsT−Ωξ
EA
5. Compute kinematic model of joint motion rate: q3 ← lsmaε(2rj)−1
6. Integrate q3
7. Return q3
limits. In step 3 the calculation of the phase transformation allows for the analysis of the
hysteresis loop when SMAs are transitioning from martensite to austenite and vice versa. In
step 4 SMA strain is calculated allowing for the estimation of elbow joint rotation rate.
4.6 Simulation and experimental results
Simulations and experiments are aimed at analyzing three issues:
1. Wing torques for actuation: It quantifies the required torques to properly move the joints
of the wing skeleton. It allows for the characterization of actuators (simulation).
2. Body torques for maneuvering : It quantifies the influence of wing inertia into the pro-
duction of rolling and pitching torques for maneuvering. It allows for the assessing of
forward and turning flight. Also, it is useful for attitude control tuning and requirements
for proper wing modulation (simulation and experimental).
3. SMA actuation limits : It explores the limits to safe overload the response of the SMA
actuators by defining the maximum value of input electrical current that achieves the
fastest rotation speed of the elbow joint (simulation).
An open-loop Matlab-based simulator has been implemented using the SimMechanics tool-
box of Simulink1. Figure 4.12 details the main modules that compose the simulator. One key
advantage of the simulator consists on the possibility of using the CAD model of the robot
exported directly from SolidWorks2. This allows to include the mechanical assembly together
kinematics and dynamics properties of the robot into the simulation environment. The sim-
ulator will be extended to a closed-loop architecture in Chapter 6.
1http://www.mathworks.es/products/simmechanics/index.html2http://www.solidworks.com/
74
4.6 Simulation and experimental results
4.6.1 Open-loop simulator
This simulator is composed by the following modules (cf. Figure 4.12):
• Wing trajectory generator : It generates trajectory patterns (qi, qi, qi) for each joint of the
wings at specified wingbeat frequency f . These patterns are similar to those shown in
Figures 4.5 and 4.5.
• Kinematics module: it places frames of references into the mechanical assembly of the
robot. It also maps the wing trajectory patterns to the corresponding wing joint. Spatial
operators for rotation and translation are calculated (cf. Annex 10.1).
• Dynamics : It contains the inertial model (cf. Algorithm 1). It calculates the required
torques (τi) to produce the desired wing motion patterns (qi, qi, qi). Also, it calculates
the spatial inertial forces (FT ) that are generated at the center of mass of the robot due
to the motion of both wings. This allows for quantifying the influence of wing inertia for
maneuvering.
• SMA actuation: It maps position commands for elbow and wrist rotation (q3) to out-
put torque (τ3). The output torque is generated by calculating the required amount of
input heating power (I2smaRsma) to drive SMA strain to desired elbow rotation (q3), cf.
Algorithm 2.
• Mechanics module: It contains the mechanical assembly and properties of robot’s CAD
exported directly from SolidWorks (cf. section 5.1 in Chapter 5 to further details on CAD
design).
4.6.2 Wing torques for actuation
To characterize the actuators for wing actuation, the inertial model in Algorithm 1 is used. It
allows for the estimation of wing torques (τi) based on desired wing trajectory profiles (qi).
Figure 4.13 describes how wing torques are applied to the bat robot. Effective forces τ1, τ2 are
the torques applied to the shoulder joint. The former allows for the primary flapping motion
(q1) whereas the latter for the forward/backward rotation of the wings about the gravity axis
(q2). The effective force τ3 is the torque applied to the elbow joint. It allows for the contraction
and extension of the wing via elbow rotation (q3). Also, τ4, τ5 and τ6 are the torques applied
to the wrist joint which rotate the digits to maintain wing membrane tension. Further details
on actuator(s) selection based on the simulation data presented in this section can be found in
75
4.6 Simulation and experimental results
Wing
traje
ctory
gen
erat
or m
odule
SMA
actu
ation
mod
ule
Kine
mat
ics, m
echa
nics,
dyna
mics
mod
ules
User
ope
ratio
n
3D e
scen
ario
Figure
4.12:Sim
Mechanicsopen
-loopsimulatorfordynamicsandSMA
actuation.Sourc
e:th
eauth
or.
76
4.6 Simulation and experimental results
q3
q4,5,6
q2
q7
xb
yb θ
φ
xb
ybτφ
τθ
z2
z1x1
y1
y2x2
z3
x3
y3
x4,5,6
y4,5,6
z4,5,6τ1
τ 2
τ 3
τ 4,5,6
Inertialmodelqi τ i
aerodynamics
robotparameters
Figure 4.13: Wing torques denoted as τi correspond to the effective forces that the each joint i
requires to rotate as defined by trajectory profiles qi (see upper inset). Torques are with respect
to the joint frames {i} assigned by modified DH convention. The lower inset shows the model for
the estimation of torques as a function of the joint profiles. The inertial model in Algorithm 1
takes into account robot parameters/constrains and aerodynamic loads. Source: the author.
Table 4.5: (forward flight) wing torques as a function of the wingbeat frequency f .
f = 1.3Hz f = 2.5Hz f = 5Hz f = 10Hz
|τ1|[Nm] 0.00016 0.0006 0.0024 0.0096
|τ2|[Nm] 0.00037 0.0013 0.0054 0.022
|τ3|[Nm] 0.0007 0.0027 0.011 0.0438
|τ4,5,6|[Nm] 0.00027 0.0010 0.004 0.0162
Chapter 5 section 5.3.1.2. For instance, the following simulation are aimed at characterizing
wing torque requirements for: i) forward flight maneuver, ii) turning flight maneuver, and iii)
energy-demanding kinematics pose.
Required torques during forward flight
Here, wing profiles qi are applied according to the kinematics trajectories defined to forward
flight in Figures 4.5. The response of the inertial model is shown in Figure 4.14. Because during
forward flight both wings flap symmetrically describing the same wing profile, the torque values
shown in Figure 4.14a apply for both left and right wings. This wing torque profile corresponds
to a wingbeat frequency of f = 5Hz, which is the typical flapping frequency during steady
flight. From the figure is easy to appreciate how the elbow joint (red plot) demands for about
77
4.6 Simulation and experimental results
0.01
0.02
0.03
0.04
0
-0.01
-0.021.3 2.5 5 10
−τ 1−−τ 2−−τ 3−−τ 4,5,6−
(a)
(b)
Figure 4.14: (Simulation) a) required wing torques (τi) during forward flight. Both wings flap
symmetrically describing the wing profiles qi shown in Figure 4.5b (wingbeat frequency f = 5Hz),
b) Increments of wingbeat frequency cause the wing torques to increase. The plot shows maximum
peak values of wing torques Source: the author.
4.5-times more applied torque than the flapping motion powered by the shoulder joint (from
2.4mNm to 11mNm). This is caused by gyroscopic forces generated during the flapping motion
which directly affect the in-plane contraction and extension of the wings. These results provide
a first insight about how much output torque the SMA actuators should produce to drive the
elbow and wrist rotations (τ3 + τ4,5,6). Also, to assess and quantified how wing torques scale
as a function of the wingbeat frequency (f), Figure 4.14b details the response of the inertial
model considering different wingbeat frequencies ranging from 1.3Hz up to 10Hz. In plot-b
the x-axis corresponds to four sets of frequencies. Each set contains the wing torques of each
joint. As f increases, note how the wing torque profile also increases due to the increment
of angular accelerations of each joint. Plot-b is useful because provides an insight about how
the wing torque profile scale up when f increases, being not only useful for the selection of
actuators but also for control tuning. An scaling factor denoted as Winertial−factor has been
quantified from the results in plot-b. Several simulation results have been carried out to confirm
that the wing torque profile scales by a factor of ∼ 4 when the wingbeat frequency is doubled
78
4.6 Simulation and experimental results
1.3 2.5 5 10 1.3 2.5 5 10
−τ 1−−τ 2−−τ 3−−τ 4,5,6−
−τ 1−−τ 2−−τ 3−−τ 4,5,6−
Wingbeat frequency f [Hz] Wingbeat frequency f [Hz]
(a) (b)
Figure 4.15: (Simulation) a) required wing torques (τi) during turning flight: a) left wing,
b) right wing. The upper plots show the difference between the wing torque profile that each
wing requires to describe the trajectory profiles qi shown in Figure 4.6b-e respectively (wingbeat
frequency f = 5Hz). The lower plots show how increments of wingbeat frequency cause the wing
torques of each wing to increase. Source: the author.
(Winertial−factor = 4). Table 4.5 reports the numerical data.
Required torques during turning flight
Here, wing profiles qi are applied according to the kinematics trajectories defined for turning
flight in Figure 4.6. The response of the inertial model is shown in Figure 4.15. Contrary to
forward flight where both wings are modulated symmetrically, turning flight maneuver requires
the robot to modulate each wing differently. The modulation profile for each wing was previously
presented in Figures 4.6b-e. To goal behind this approach is to generate body torques at the
center of mass of the robot that result on producing body angular accelerations that allow for
turning left or right. In this simulation the left wing has been modulated with a morphing wing
factor of 0.1 which corresponds to the kinematics profile shown in Figures 4.6a. This causes the
left wing to be more contracted with respect to the right one. This causes the center of mass to
displace towards the right wing and therefore generating rolling torque around the body frame
{b} (cf. 4.16b, lower plot).
This approach works a shown in Figure 4.16a. Up to t = 2.5s both wings are modulated
symmetrically with the same wing trajectory profile qi, which causes net inertial forces to cancel
79
4.6 Simulation and experimental results
..
..
..
yb
xbφ
(a) (b)
Figure 4.16: (Simulation) effects of wing modulation on the production of body torques and
angular accelerations at wingbeat frequency of f = 1.3Hz a) wing modulation profile qi for the
left wing (upper) and right wing (lower), b) (upper) body torques (τθ, τφ, τψ) produced at center
of mass, (medium) body angular accelerations (θ, φ, ψ) produced at center of mass, and (lower)
attitude response of the robot (θ, φ, ψ). Values are with respect to the body frame {b}, Source:the author.
each other at the center of mass of the body, cf. Figure 4.16b (upper plot). As a consequence zero
body accelerations in roll (φ) and yaw (ψ) are produced, cf. Figure 4.16b (medium plot). Now
from t > 2.5s, the right wing reduces the amplitude of the joint modulation by a morphing-wing
factor of fmc = 0.5 (2.5 < t ≤ 3.8) and then with fmc = 0.1 (t > 3.8), cf. Figure 4.16a (lower
plot). As a consequence rolling and yawing torques are being produced, causing a dominant
body angular acceleration in roll that makes the body to rotate towards the expanded left wing.
It is important to mention that body torque components in Figure 4.16b (upper plot) have been
extracted from the spatial body forces FT calculated by the inertial model using Algorithm 1.
Likewise body angular accelerations in Figure 4.16b (medium plot) have been extracted from
the spatial body accelerations Ab, whereas attitude response in Figure 4.16b (lower plot) is
calculated from the double integration of angular accelerations with respect to time. Further
simulations in Figure 4.17 show how body torques (upper plots), body acceleration (medium
plots) and attitude response (lower plots) are increased when the wingbeat frequency increases
up to f = 10Hz.
Also, Table 4.6 details numerical values for the characterization of torque requirements for
each joint of the wing qi. It specifies how much torque should be applied depending on the
80
4.6 Simulation and experimental results
..
..
..
..
..
..
..
..
..
0.2
0.1
0
-0.1
0.15 0.2
0-0.20
-0.150
(a) (b) (c)
Figure 4.17: (Simulation) effects of wing modulation on the production of body torques and
angular accelerations at wingbeat frequency of a) f = 2.5Hz, b) f = 5Hz, and c) f = 10Hz.
(upper) body torques,(medium) body angular accelerations, (lower) attitude response. Source:
the author.
Table 4.6: Characterization of actuators: wing torque requirements for flapping at 10Hz and
morphing1 at 2.5Hz
Parameter q1 q2 q3 q4,5,6
Joint rotation range2 qi[deg] 120 25 60 45, 30, 18
Joint speed q[deg/ms] 120/50 25/50 60/200 45, 30, 18/200
Torque3 τi[Nm] 0.0096 0.022 0.0027 0.001
Torque (±5% tolerance) τi[Nm] 0.010 0.023 0.0028 0.00105
1 Morphing motion is slower due to SMA limitations, cf section4.6.4.2 Joint rotation ranges applied during forward flight, cf. Figure 4.5b.3 It contains aerodynamics loads experimentally quantified in Figure 4.10.
specific joint speed and its kinematics range. This characterization will be useful for the selection
of the flapping motor and also for the selection of SMA morphing-wing actuators. The following
section details how to properly generate rolling and pitching torques for maneuvering based on
changing wing morphology. It allows for the experimental quantification of the influence of
wing inertia on the production of body forces and accelerations. Also, simulation results are
contrasted against experimental measurements of body forces aimed at validating the inertial
model in Algorithm 1.
81
4.6 Simulation and experimental results
4.6.3 Body torques for maneuvering: experiments for inertial model
validation
Back in section 4.4.2 it was explained that net forces Fnet (cf. Eq. (4.20)) varies during the
wingbeat cycle in part due to the inertial thrust (forward-oriented component: FT,x, cf. Figure
4.9) changes from upstroke to downstroke and vice versa. During downstroke, the downward
and forward acceleration of the wings produce an inertial force that moves the body backward
and upward, which produce negative inertial thrust. On the contrary, the upstroke motion
produces positive inertial thrust mainly due to the body moves in opposition to the flapping
direction in order to conserve momentum. This issue has been simulated and experimentally
proven in Figure 4.18.
Figure 4.18a details the end of the upstroke motion whereas Figure 4.18b the ends of the
downstroke motion. Upper plots show experiments carried out to measure 6D forces that are
generated by the wings during the wingbeat cycle. The same testbed used in aerodynamics
experiments from Figure 4.10 was used for inertial measurements1. Lower plots show simulation
results using the SimMechanics simulator shown in Figure 4.12. As mentioned, the simulator
contains the inertial model from Algorithm 1 that allows for the computation of inertial forces
FT and net forces Fnet. Firstly it can be noted from Figure 4.18 (lower plots) how the primary
flapping motion of the wing generates forward and backward pitch motion (θ) measured about
the body frame {b} and with respect to the inertial frame {o}. Pitch motion is generated by
the pitching torque (τθ) produced when the wings are placed forward and backward the body.
Quantification of pitching torque (τθ) and also rolling torque (τφ) are consigned in Figure 4.19.
To generate forward pitching torque (τθ) both wings must be positioned towards the body
(see inset from Fig. 4.19a). The blue plot corresponds to the experimental values of τθ measured
by the force sensor, whereas the black plot corresponds to the response of the inertial model
in Algorithm 1. The simulated pitching torque has been extracted from the spatial force FT
as defined in Eq. (4.17). The disparity in results between the simulation model and the
experimental results are due to two factors. Firstly, mechanical asymmetries due to small
fabrication errors have caused the robot to generate a small component of pitching torque
(τθ =∼ 0.06Nm) even when q2 = 0o (from t = 0s to t = 2s in Fig. 4.19a). The solution
of this problem requires the incorporation of an attitude controller that regulates the pitch
motion. Secondly, the disparity in the amplitude of the oscillations is due to non-modeled
1force sensor Nano17 transducer ATI Industrial Automation, 0.318 gram-force of resolution, http://www.
ati-ia.com/
82
4.6 Simulation and experimental results
UpstrokeDowntroke
zb
xb
yb
{b} zb
xb FT,x
zb
xb
FT,x
(a)
(b)
force sensor
CMCM
Upstroke Downtroke
(a)
Figure 4.18: Effects of the flapping motion of the wings on the position of the center of mass (CM)
and accelerations of the body: a) end of upstroke motion, b) end of downstroke motion. Upper
plots depicts the testbed for experimental measurements of six-dimensional inertial forces (FT )
and lower plots depicts simulations for the computation of inertial forces (FT ). Values for both
measured and simulated FT are consigned in following Figure 4.19. To conserve momentum, the
body moves in opposition to the flapping direction. During upstroke, the upward and backward
acceleration caused by the flapping motions of the wings produce an inertial force (red circled
arrow) that moves the body forward and downward with respect to the downstroke. This force
produces a forward-oriented component (FT,x), or inertial thrust (green solid arrow). Contrary,
during the downstroke negative inertial thrust is produced. Source: the author.
physical quantities, such as mechanical frictions. This issue may cause the simulation values to
be larger in amplitude in comparison to the experimental readings.
To generate rolling torque (τφ) that allows the robot for turning left or right one wing must
be contracted while the other is extended (see inset from Fig. 4.19b). In this experiment similar
disparity problems to those observed from the pitch test scenario are presented. However, note
from both Fig. 4.19a-b that from t = 2s to t = 4s the model and the experimental values tend
to stabilize about the same bias value of torque. In plot-a the bias value of τθ =∼ 0.31Nm,
whereas in plot-b the bias value of τφ =∼ 0.11Nm.
Figure 4.19c-d shows how bias values of τθ and τφ scale up when the wingbeat frequency
83
4.6 Simulation and experimental results
[Nm]
2 [deg]
[Nm]
3 [deg]
qyb
(a) (b)
(c) (d)
1.3Hz 2.5Hz5Hz
1.3Hz 2.5Hz5Hz
xb
q
xb
yb
q
xb
yb
q3 (deg)q deg
t (s) t (s)
τ θ(Nm)
τ θ(Nm)
τ φ(Nm)
τ φ(Nm)
q deg q deg, q deg
q deg, q deg
q
xb
yb
Figure 4.19: (Experimental) wing inertia contribution on forward and turning flight: a) sim-
ulation model VS experimental results of pitching torques τθ (f=5Hz), b) simulation model VS
experimental results of rolling torques τφ (f=5Hz), c)-d) quantification of wing inertia contribution
into the generation of τθ and τφ at different wingbeat frequencies f . Source: the author.
increases. Table 4.7 details the numerical results of the scaling factors for both pitching and
rolling torques. It has been observed that the influence of wing and body mass (mt = 0.125Kg)
on the production of inertial torques is proportional to m1/ft , ∀f > 0. Biological studies in
(24) show experimental evidence about the implication of this relationship in terms of inertial
and aerodynamics contribution. For the application at hand, mt has accounted for ∼ 50% of
pitching torque production, and ∼ 20% for rolling torque production. By adding the membrane
to the wings, the rest is contribution of aerodynamic forces.
Other interesting tests in Figures 4.20 and 4.21 show how the production of pitching and
rolling torques respectively is affected when considering non-biologically inspired wing joint
references but instead step-input signals that drive wing joint rotations.
The set M1 corresponds to the inertial torques generated by using the biological trajectory
84
4.6 Simulation and experimental results
Table 4.7: Wing and body mass influence in the generation of τθ and τφ (mt = 0.125kg)
Scaling factor f = 1.3Hz f = 2.5Hz f = 5Hz
Pitch1(0.5m1/ft )
Experimental 0.11Nm 0.2Nm 0.31Nm
Simulation 0.1Nm 0.21Nm 0.33Nm
Roll2(0.2m1/ft )
Experimental 0.04Nm 0.07Nm 0.11Nm
Simulation 0.04Nm 0.08Nm 0.13Nm
1 Values from Fig. 4.19c2 Values from Fig. 4.19d
y
x
Figure 4.20: (Simulation) pitching torque response (τθ) for different positions of q2 (wings rotated
forward/backward the body) and for different wingbeat frequencies f . The set M1 contains bio-
inspired wing modulation shown in Figure 4.5b whereas set M2 corresponds to step-input signals
that rotate each joint of the wing due to the range defined in Table 4.6 (cf. joint rotation range).
Source: the author.
patterns, whereas the set M2 using step-commands patters. In the first place one can note a
significant differences between the torques values obtained from both simulations. Table 4.8
details the numerical results. The kinematics modulation of the wing joints has a key impact into
the production of rolling and pitching torques for body rotation. It is precisely the geometry of
the wings that causes different inertial and aerodynamic responses depending of the modulation
profile that changes its shape during flapping. In most of the cases, note how the percentage
of torque increase is greater than 50%, which conclusively stands for the importance of using
bio-inspired modulation profiles presented in Figures 4.5 and 4.6. By including control methods
that ensure wings are always modulated with the proper bio-inspired patterns, section 7.2.1 in
Chapter 7 will introduce further experimental results about the importance and influence of wing
inertia modulation for more efficient flight.
85
4.6 Simulation and experimental results
y
x
Figure 4.21: (Simulation) rolling torque response (τφ) for different positions of q2 and wingbeat
frequencies f . The set M1 contains bio-inspired wing modulation whereas set M2 corresponds to
step-input signals that rotate each joint of the wing. Source: the author.
Table 4.8: Advantage of using bio-inspired wing joint trajectories on the production of pitching
and rolling torques).
Trajectory set f = 1.3Hz f = 2.5Hz f = 5Hz 10Hz
τθ[Nm] (Pitching1)
M2 0.08 0.15 0.2 0.3
M1 0.11 0.2 0.35 0.55
% of increase 37% 34% 75% 84%
τφ[Nm] (Rolling2)
M2 0.01 0.025 0.06 0.09
M1 0.02 0.04 0.09 0.15
% of increase 100% 60% 50% 67%
1 Values from Figures 4.20 for q2 = 20o
2 Values from Figures 4.21 for q2 = −20o
4.6.4 SMA-muscle limitations
This section presents simulation results that allow for the assessment of SMA limitations in
terms of: i) actuation speed, and ii) power consumption. Both issues are fundamental for
tackling the SMA control problem aimed at driving morphing-wing motions. Using the SMA
phenomenological model previously introduced in Algorithm 2, simulations are focused on ex-
amining how an SMA wire respond upon high values of input current (Isma). This allows for
the definition of input power limits to overload SMA nominal operation. Therefore nominal and
overloaded SMA operation are evaluated. Nominal operation mode is defined by the manufac-
86
4.6 Simulation and experimental results
0 0.1 0.2 0.3 0.4 0.50
7
14
21
28
35
t[s]
[deg
]
0 0.1 0.2 0.3 0.4 0.5
50
100
150
200
t[s]
T[de
g]
0 20 40 600
50
100
200
0.02 0.03 0.04 0.05 0.06 0.0750
100
150
200
[MPa
]
@350mA@215mA@175mA@1A
@215mA@175mA@350mA@1A
Lower Nominal Overloaded Overheated
1
1
2
2 3
34
4
3
(d)
T [°
C]
cooling
heating350mA
ISMA =550mA
0
15
30
45
60
75
ISMA =350mAISMA =175mAISMA =1A
q3 [d
eg]
t [s]
(a)
[MPa
]
I=350mAI=175mAI=550mAI=1A
0.04 0.03 0.02 0.0150
200
heating
350mAI =350mA(b)
50454035302520
20
40
60
80
100T [C]
% A
uste
nite
20 25 30 35 40 45 5020
40
60
80
100
% M
artensite
Cooling
HeatingMs
Af
AS
Mf
(c)
Nominal(350mA)
(a)
T [°C]
t [s]
25 30 35 40 4520 5020
40
60
80
100
150
100
20
40
60
80
100
ISMA =350mAISMA =175mAISMA =550mAISMA =1A
Figure 4.22: (Simulation) SMA phenomenological model response at different current profiles:
a) Joint rotation based on SMA strain. b) Temperatures on the SMA wire, c) Hysteresis-loop for
the nominal operation mode (Isma = 350mA), d) SMA strain VS stress.
turer of the NiTi SMA wire (Migamotor, http://store.migamotors.com/, nanomuscle-nm706
super linear muscle wire actuator), cf. section 5.3.4 in Chapter 5 to further details on SMA
actuators supplied by Migamotors. This muscle-like actuator consists of several short strips of
SMA wire attached to opposite ends of six metal strips stacked in parallel. Each SMA segment
pulls the next strip about 0.67mm relative to the previous strip, and the relative movements
sum to make a stroke of 4mm. The SMA mechanical configuration shown in Figure 4.11b allows
an elbow rotation range of q3 = 60o (the radius of the elbow joint is rj = 0.004m).
In summary, under nominal operation the SMA wires have a strain of 4% and are able to
contract in 300ms while pulling 12.23gram − cm (torque at joint of 0.0012Nm) by applying
an input power of 0.26W (Isma = 175mA) at 3V . In this thesis alloys of 150μm of diameter,
with length of Lsma = 0.035(×6)m and nominal electrical resistance of Rsma = 8.5Ω are used.
SMA parameters are described in Table 4.4. On the other hand, overloaded mode refers to
increases in input electrical currents that drive faster joint rotation rates. Because using larger
input currents may cause physical damage of the shape memory effect, simulation analysis is
convenient.
Figure 4.22 shows the response of the phenomenological model (cf. Algorithm 2) under
different values of input heating currents Isma, ranging from 175mA up to 1A. Figure 4.22a
87
4.6 Simulation and experimental results
shows the angular rotation profiles (e.g. elbow’s joint q3) obtained for each input current.
Figure 4.22b shows increases on SMA wire temperature (T ) corresponding to the results shown
in plot-a (cooling temperatures are detailed in the inset). Figure 4.22c details the hysteresis loop
of the SMA wire at input current of Isma = 350mA (states from Austenite to Martensite and
viceversa). Finally, Figure 4.22d depicts the strain (ε) versus stress (σ) curves corresponding
to the results shown in plot-b.
To assess how fast the elbow joint can rotate the kinematics model from Eq. (4.26) is used.
It relates the strain rate of the SMA wire (ε) with the rotation speed of the elbow joint (q3).
As a first attempt, an input current of 175mA is applied, being this value, the nominal input
current suggested by the manufacturer of the SMA actuators (96). Note that under this value,
the joint rotates ∼ 60o in 400ms, resulting too slow for the application at hand. As expected,
by increasing the input current up to 1A, faster angular motions were achieved (up to ∼ 60o in
75ms), however, the SMA wire finish austenite temperature (see Figure 4.22b) was dramatically
increased up to Af = 150oC, being this value about twice higher than the upper value defined
for the simulation Af = 78oC (safe temperature limit of the alloy, see Table 4.4). This issue
clearly illustrates an overheating problem. Overheating causes an increase in the cooling time
of the SMAs, which makes the actuators slower. It also can cause physical damage to the shape
memory effect.
In order to classify the SMA’s operation modes upon the responses of input heating currents
Isma, four modes of SMA operation have been defined: 1-lower, 2-Nominal, 3-Overloaded, 4-
Overheated/overstressed. Both nominal and overloaded modes are feasible targets to pursuit
with the experiment, e.g., by applying an input current around 550mA, the joint is able to
rotate 60o in 100ms while keeping the maximum limits of temperature and stress below the
limits of overheating, i.e., above the limits of the finish austenite temperature Af .
As long as the input heating current Isma is smaller than a maximum allowed current Imax,
the SMAs will be safe, and the angular speed resulting from this input current could be set as
a feasible target speed to pursuit in the experiment. For instance, simulation results suggest
Imax =∼ 550mA. Table 4.9 summarizes numerical values to safe overload SMA operation.
Nominal mode allows for a morphing-wing frequency of 1.3Hz whereas overloaded mode allows
for a morphing-wing frequency of 2.5Hz. These values are defined by the simulation results of
elbow joint angular speed carried out in Figure 4.22a.
88
4.7 Remarks
Table 4.9: Limits for overloading SMA operation
Parameter Theoretical1 Simulation2
Nominal (f = 1.3Hz)
Max. input current Isma [mA] 175 350
Max. input heating power uheating [W ] 0.26 1.04
Resultant joint speed q3 [deg/ms] 60/300 60/200
Resultant output torque τ3 [Nm] 0.0012 0.0007
Overloaded (f = 2.5Hz)
Max. input current Isma [mA] N/A 550
Max. input heating power uheating [W ] N/A 2.57
Resultant joint speed q3 N/A 60/100
Resultant output torque τ3 [Nm] N/A 0.0027
1 Nominal values provided by Migamotor’s model NanoMuscle RS-70-CE 1131 (96). Values do not consider external
load. Overloaded values are not provided.2 Simulation results from Figures 4.22a (resultant joint speed at input current), and Table 4.5 (resultant output
torques). Values do consider the wing-skeleton load.
4.7 Remarks
This chapter has presented the formulation of kinematics, dynamics, aerodynamics and SMA
actuation via modeling frameworks that are a key step towards the design/fabrication, control,
and flight experiments using BaTboT.
The most important contributions of this chapter can be summarized as follows:
• Flight maneuvers were kinematically formulated in Figures 4.5 and 4.6. It has shown how
the bio-inspired kinematics of wing joint modulation has allowed the definition of forward
and turning flight. As a result, wing joint trajectory profiles were determined.
• An inertial model formulated in Algorithm 1 has allowed the assessment and quantifica-
tion about how wing inertia affects robot’s maneuverability, and the key role of proper
wing modulation aimed at the production of rolling and pitching torques for turning
and forward flight. It has also allowed the characterization of torques requirements for
actuator selection.
• A phenomenological model formulated in Algorithm 2 has allowed the assessment and
quantification of limits to overload SMA operation by preventing the wires to overheat
and overstress.
The following chapter introduces the design and fabrication process of BaTboT and its
components.
89
5
BaTboT design and Fabrication
The rich diversity of mechanisms of animal flight can provide abundant inspiration for
engineered design.
5.1 The general method for BaTboT’s design
Flapping flight is the single most evolutionarily successful mode of animal locomotion: there are
today over 1200 species of bats, more than 10000 living species of flying birds, and somewhere
between millions and tens of millions of species of flying insects. Understanding how animals fly
is not only central to providing insight into the biological world; the rich diversity of mechanisms
of animal flight can provide abundant inspiration for engineered design.
This chapter introduces the design and fabrication process of BaTboT, the first bat-like
micro aerial vehicle with highly articulated wings. This chapter is not intended to cover
performance of the mechanical designs presented, but only the design process,
main functions and the criteria involved. Refer to Chapters 4 (modelling) and 6
(Control) for detailed experiments regarding the mechanical approaches introduced
herein. In this chapter, one of the most important challenges to be tackled concerned to
the design criteria to specify morphological and actuation parameters. Because the novelty
of the robot and the lack of information regarding design issues, this thesis was meant to
design BaTboT from the analysis of a specific bat specimen aimed at carefully mimicking each
detail related to morphology, biomechanics, kinematics and even the muscle-like wing actuation
system. Chapter 3 already presented this biological analysis by concluding the chapter with
key issues or foundations to the design process. Here, those foundations are bring back and
incorporated into the design framework that brings BaTboT to live.
90
5.1 The general method for BaTboT’s design
wing markers
1. Biological study: kinematics extraction of
wings' motions
2. Biological inspiration:Morphology, physiology
and design criteria
3. Fixed-wing design:Flapping testbed and membrane
testing. Flapping actuation selectionElbowjoint
Carpus/wristjoint
MCP-III digitshoulder
4. Articulated-wing design:Metal tendons provide wrist
motion
air-bubbles within the membrane detected!!
SMA-based muscles
5. Morphing-wing mechanism:Antagonistic design of SMA-based muscles as actuators.
Plagiopatagium skin(0.1mm silicon wing
membrane)
R/C transmission link49MHz
Antenna
motor+electronics
SMAmorphing muscles
6. Proper membrane design and testing:inertial force measurements of joint
torques.7. Robot assembly + electronics:
control encoded.
9. Flight tests
Re-designing process and adjustments
8. Wind-tunnel measurements.Wind-tunnel testing of inertial and
aerodynamics forces. Control response testing.
Figure 5.1: Design process of BaTboT. Source: The author.
91
5.2 Prototype characteristics
5.2 Prototype characteristics
This section briefly gives an insight about the design process previously mentioned and the
main components of the robot. The following sections describe seven major steps that are key
into the design process of the robot.
5.2.1 Design process
The design process is depicted in Figure 5.1. Steps 1 and 2 highlight the key biological founda-
tions concluded from Chapter 3, section 3.3.2. It details how biological based parameters are
used for the CAD design of the robot. Step 3 shows the first approaches for the design and
fabrication of the wing skeleton and the membrane. It details the first problems and solutions
to transform a rigid skeleton into a articulated wing. Step 4 and 5 present the final design of
the articulated wing that incorporate the muscle-like actuation system using Shape Memory
Alloys. In step 6, the final design and fabrication process of the wing membrane is presented
and attached to the wing skeleton. Experiments are conducted for testing the elasticity and
cambering properties of the silicone-based membrane. Finally, step 7 shows the components of
the robot, from electronics and sensors to the final assembly of BaTboT.
5.2.2 Components and weight distribution
Figure 5.2 details the most relevant components of the robot and their weight distribution.
Components are classified into two categories: i) structural components such as the body,
shoulder, wings and ii) electronic components such as the actuators, arduino-board, battery
and sensors. Structural components have been designed using SolidWorks CAD and fabricated
via ABS (Acrylonitrile butadiene styrene) 3D printing process. All components or parts of the
robot are hollow plastic structures that minimise the overall weight. Structural components
are shown in Table 5.1.
Wing components (i.e, shoulder, humerus, radius, digits and wing membrane) have been
carefully designed based on morphological parameters analysed from the specimen Pteropus
poliocephalu. In general, BaTboT is about 50% smaller than the true specimen’s size, but it
maintains the biological proportions of body and wing size. This is because the wing-to-body
mass ratio play an important role for maneuvering aspects that have to be included in the
flight control. Because in nature bats modulate solely wing inertia to produce pitch and roll
moments, wing mass is crucial. As shown in Table 5.2 the specimen has a wing-to-body mass
92
5.2 Prototype characteristics
Monday, July 9, 12
humerusbone
leg
shoulder
MPC-III
radiusbone
IV
V
body
arduinonano
IMU
LiPobatery
SMA driver
SMA muscles
spring bias
motor
elbowjoint
wristjoint
(a)
(b)
[grams]
Figure 5.2: a) weight distribution of main components, b) detailed view of main components.
Source: The author.
ratio of 0.404, whereas the robot has a wing-to-body mass ratio of 0.410. This closer correlation
enables high levels of performance in terms of pitching and rolling torque production.
93
5.2 Prototype characteristics
Table 5.1: Main structural components of BaTboT. CAD design
CAD Description
Body. It is the main structure of the robot with a mass of
38g. Body width is 7.8cm, a required length to achieve the
angle range of motion of the wings from the rotation axis of
the flapping DC-motor. In the middle of the body is located
the flapping motor-case (in grey colour), which consists of a
reinforced structure that holds the motor. The reinforcement
is because the larger torsional forces produced during flapping.
The size of the structure has been set to allow the motor an
specific x, y, z position to drive the flapping of the wings. In
addition the structure contains an extra space for the second
axis of rotation. Section 5.3 details on the requirements of the
flapping-wing mechanism. All the compartments inside the
body work as hosts for the electronic components, as shown
in Figure 5.1b.
.
Shoulder. It is a structure with a mass of 3g and carefully
designed to fit in the body. The robot has two shoulders
to drive the flapping motion of each wing. Both wings are
attached to each shoulder which provides the corresponding
angle of attack to each of them (AoA = 9o). Section 5.3
details the flapping-wing mechanism.
.
Humerus bone. It is a structure with a mass of 4g and a length
of 5.5cm. This artificial bone connects the shoulder joint with
the elbow joint and hosts the SMA muscle-like actuators that
act as biceps and triceps. In the specimen this bone has a
length of 11cm (cf. Table 5.2). The artificial humerus is al-
most half the size of the biological one. The idea behind this
choice concerns to reducing the size of the robot’s wingspan
to the half aimed at fabricating a small-scale vehicle.
.
Radius bone. It is a structure with a mass of 2g and a length
of 7cm. This artificial bone connects the elbow joint with the
wrist joint. Inside the bone, metal tendons are extended to
allow for the motion of the wrist joint. Subsection 5.3.4 details
the morphing-wing mechanism. Also, this bone is about half
the size of the the biological one (cf. Table 5.2).
.
Digits MCP-III,IV,V. Three artificial digits with a respec-
tive mass of 0.3, 0.2, 0, 2g complement BaTboT’s wings. The
MCP-V acts as fifth digit and thumb. The thumb section has
a length of 1.76cm and it is cambered 5o. This contributes in
cambering the attached membrane. Also, the MCP-III is com-
posed by two sections, one acting as second digit; the shortest
with 5.42cm and the third digit; the longest with 10.54cm.
Digits are all connected to form the wrist joint that rotates
(open and close) as a function of the elbow joint. This sub-
actuation is driven by the metal tendons.
.
94
5.3 BaTboT mechanics
Table 5.2: Comparison of morphological parameters between the specimen and BaTboT
Parameter [unit] Bat 1 Robot
Total mass mt [g] 98 125 (with battery)
Extended wings’ length B [m] 0.462 0.245
Body width lm [m] 0.07 0.04
Body mass mb [g] 41 38
Wing mass2 mw [g] 16.58 15.6
Wing-to-body mass ratio mw/mb 0.404 0.410
Body inertia tensor diagonal Ib [gcm2] – [1, 0.07, 0]
Extended wings’ span S = lm + 2B [m] 0.99 0.53
Extended wing’s area Ab [m2] 0.069 0.05
Humerus length lh [m] 0.11 0.055
Humerus average diameter 2rh [m] 0.0055 0.006
Humerus inertia tensor diag. Jh,cm [gcm2] – [0.03, 0.37, 0.93]
Humerus position vector to CM sh,cm [m] – [0.0275, 0, 0]
Radius length lr [m] 0.145 0.070
Radius average diameter 2rr [m] 0.0042 0.005
Radius inertia tensor diag. Jr,cm [gcm2] – [0.07, 0.92, 0.37]
Radius position vector to CM sr,cm [m] – [0.035, 0, 0]
Plagiopatagium membrane thickness [m] 0.0002 0.0001
1 Morphological parameters of the specimen extracted from (22).2 wing mass is composed by: humerus (4x2g), radius (2x2g), MCP-III,IV,V (0.7x2g), SMA (1.1x2g).
5.3 BaTboT mechanics
This section details the seven steps mentioned in the Design Process flow depicted in Figure 5.1.
Step 1: Biological study was already carried out in Chapter 3: From bats to BaTboT: Mimicking
biology. Data extracted from in-vivo experiments presented from the biological literature was
analysed and used as a framework for the i) selection of the specie to mimic and ii) BaTboT
design. Regarding the first point, several species were evaluated in terms of flight performance
and morphological characteristics. The evaluation focused on the Rousettus aegyptiacus speci-
men, with similar physiological and aerodynamical behaviour than the Pteropus poliocephalu.
Kinematics, dynamics and aerodynamics data was then compared. Regarding the second point,
Chapter 3 was concluded with key bio-inspired geometrical parameters for modeling and design
(cf. Table 3.5) and a biological-based framework for design aimed at achieving the best flight
performance found in nature (cf. Table 3.6). In the following, both criteria are consolidated for
the final design parameters of BaTboT.
95
5.3 BaTboT mechanics
humerus bone
radius bonethumb
leg
elbow joint
wrist joint
SMA musclesbiceps, triceps
shoulder joint
wing membrane
MCP-III
MCP-IV
MCP-V
(a)
(b)
Figure 5.3: a) Each morphological aspect of BaTboT design has been carefully approached to
its biological counterpart, b) Cartoon of typical wingtip trajectory described during a wingbeat
cycle. Source: The author.
5.3.1 Step 2: Design criteria
This subsection reviews key criteria, from BaTboT design to final fabrication. For design,
bio-inspired parameters found in the specimen are used, whereas for fabrication, the inertial
model of the robot is used. The inertial model (inverse/forward dynamics model) is useful for
the estimation of actuators and general performance. The following subsection details on both
issues.
5.3.1.1 Bio-inspired parameters
In Figure 5.3a, it can be really appreciated how BaTboT design has been carefully conceived
aimed at mimicking as close as possible the morphology of the biological counterpart. Each arti-
ficial bone and skeleton component (cf. Table 5.1) has been shaped and dimensioned according
to the morphological data studied and extracted from the specimen.
As mentioned before, the robot is in general half the size of the bat, however,
skeletal proportions in terms of lengths and weights have been attempted to
maintain. This is extremely important since most of the dynamics and
aerodynamics performance relies on these proportions, specially in the ratio
between wing size/mass and body mass.
96
5.3 BaTboT mechanics
Morphological parameters are completely described in Table 5.2, whereas efficient param-
eters for design as a function of body and wing mass (mt) are presented in Table 3.6. In the
following, these parameters are highlighted, explaining how they affect dynamics and aerody-
namics behaviour. Dynamics is quantified in terms of rolling (τφ) and pitching (τθ) torque
production, which are key variables that enable efficient manoeuvrability. Aerodynamics is
quantified in terms of lift and drag production.
• Wingspan: it defines the area of the wings, larger wingspans induce larger lift forces but
also produces larger drag forces. Because BaTboT is able to modulate its wings (change
the area), minimum and maximum wingspan is defined as 0.41 − 0.46m, generating a
maximum wing area of 0.064m2. BaTboT wings have to be massive aimed at accounting
sufficient inertial forces to maneuver. However heavy wings might constraint the robot to
properly flight. The tradeoff between wing size and weight is defined as the ratio between
wing mass and wing-length. It has been observed from the specimen this ratio is about
0.35g/cm. Because BaTboT has the half of the wingspan, wing-mass to wing-length ratio
is about twice the biological one: 0.65g/cm. This difference is partly compensated with
the total mass of the robot.
• Wing-to-body mass: this ratio is crucial for the proper impact of wing inertia into
the production of body accelerations. In the specimen this ratio is 0.404 and BaTboT
has a wing-to-body mass of 0.410. This implies a correlation of 98.51%. This criterion
is perhaps the most important and influential for robot design, since the hy-
pothesis of this thesis is related to the quantification of wing inertia and its
key influence in robot’s manoeuvrability and net force production. To achieve
this closer correlation in the wing-to-body mass ratio between the specimen and BaTboT,
both body and wing skeleton of the robot was carefully shaped in the CAD (SolidWorks)
to finally match the desired mass (ABS plastic properties). Refer to Table 4.7 and Figures
4.19, 4.20 to observe how wing inertia influence was quantified in both simulation and
experimental testing.
• Kinematics of flight: it is mainly represented by flapping frequency (f). From Table 3.6
it has been observed that biological bats do not decrease the flapping frequency lower than
1.47Hz (wingbeat period of 0.68s). In addition, wings must have a wingstroke amplitude,
or flapping motion amplitude of 157.8o and a minimum angle of attack of ∼ 7o. Both
issues drastically influence the production of aerodynamic forces.
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5.3 BaTboT mechanics
• Aerodynamics of flight: flapping motion amplitude and wing’s angle of attack are not
the only kinematic variables that influence on aerodynamics behaviour. Wing camber
is also important. No-cambering constraints the proper flow of air through the wing,
whereas excessive cambering generates turbulences that increase drag and decrease the
lifting surface. Maximum wing camber is set to 0.15. Also, aerodynamics is affected by
the way wings are modulated, in other words, how to properly control the morphing-wings
to achieve the higher lift-to-drag ratio.
As shown in Table 3.6, the previous parameters vary depending on the total mass (mt). For
the application at hand, mt = 0.125Kg (including battery). In case of fabricating a bigger or
smaller robot, the design framework introduced in Table 3.6 determines how to design the most
important components of the robot that enable the kinematics, dynamics and aerodynamics
behaviour found in nature. Chapter 7 will assess and demonstrate via direct experimentation
whether these design parameters really induce the high performance expected.
5.3.1.2 Actuators
BaTboT features a hybrid drive, partially actuated by a DC motor which drives the primary
flapping motion and SMA actuators which drive the morphing-wing motion. Both actuators
have been selected based on the simulation results given by the computation of the inertial
model. Table 4.6 contains the numerical data. From the table, the joints of interest are: q1
(flapping) and q3 (morphing). In the practise, the other joints are under-actuated, which means
they have no direct actuation. Table 5.3 summarises the torque requirement criteria for the
proper selection of both flapping and morphing-wing actuators.
The flapping-wing actuator is a miniature gearmotor with dimensions (0.94” x 0.39” x
0.47”), high-power brushed DC motor with 50 : 1 gear ratio and 3mm-diameter D-shaped
output shaft. It has a mass of 9g, a rotation speed of 625RPM at 6V and 100mA free-run. It
supports a maximum current of 1.6A stall with an output torque of 1.1kg − cm. This brushed
DC gearmotor is intended for use at 6V , though in general, this kind of motor can run at
voltages above and below the nominal voltage, so it should comfortably operate in the 3− 9V
range (rotation can start at voltages as low as 0.5V ). Lower voltages might not be practical,
and higher voltages could start negatively affecting the life of the motor. Later in subsection
5.3.2.1, the response of the motor is characterised for the application at hand (flapping-wing
mechanism).
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5.3 BaTboT mechanics
Table 5.3: Actuators
Actuator Criteria3 Datasheet
Flapping1 Torque: 0.010Nm,
speed: 120/50(deg/ms)
(f = 10Hz)
Torque: 1.1kg − cm (0.1078Nm),
speed: 625RPM (f = 10.41Hz)
Morphing2 Torque: 0.0028Nm,
speed: 60/200(deg/ms)
(f = 2.5Hz)
Torque4 : (0.0012Nm), speed4 :
60/300(deg/ms) (f = 1.6Hz)
1 http://www.pololu.com/catalog/product/998.2 http://store.migamotors.com.3 Torque requirements have ±5% tolerance and contain aerodynamic loads quantified in Figure 4.10.4 Torque and speed values of SMA migamotor actuators correspond to their nominal response. They have to be
overloaded aimed at matching the requirements.
(a)
(b)
Figure 5.4: a) Pin-out and Circuit Diagrams for SMA migamotor actuator model NM706-Super,
b) actuator dimensions. Source: The author.
The morphing-wing actuator consists of several short strips of Shape Memory Alloy (SMA)
wire attached to opposite ends of six metal strips stacked in parallel. Each SMA segment pulls
the next strip about 0.67mm relative to the previous strip, and the relative movements sum to
make a stroke of 4mm. Figure 5.4 details the circuit connection diagram and the dimensions of
the actuator in both extended and contracted states. Table 5.4 details the pinout connections
of the SMA actuator.
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5.3 BaTboT mechanics
Table 5.4: Circuit pinout data for SMA migamotor actuator.
Terminal Description I/O
EXTENDED signal indicating status of full SMA extension O
CONTRACTED signal indicating status of full SMA contraction O
CTRL current signal controlling contraction I
VDC Applied voltage I
GND ground –
The maximum applied voltage VDC is 20V . When CTRL> 2.5V , the actuator contracts and
the CONTRACTED line is pulled to VDC . If an external return force, such as an antagonistic
SMA or spring produced an external pull force, the EXTENDED signal is pulled to VDC and
the unit is prepare to contract once again. For the application at hand, pull-down resistors
are necessary because the EXTENDED and CONTRACTED signals are monitored by the
arduino-board. The actuator supports driven electrical currents up to 2A, however, applying
larger currents during continuos operation may cause overheating problems, fatigue issues or
even physical damage of the shape memory effect. Driven currents have always to be monitored.
The actuator optimal operates in ambient temperatures ranging from −70oC to +75oC and
it has a cycle life of 106+. Datasheet can be downloaded at http://www.migamotors.com/
Downloads.html.
One of the key features of SMA actuators rely on their capability to act like sensors. The
inner SMA wires have an electrical resistance of 8.5Ω that decreases when temperature on the
wire increases. Several properties of the SMAs change as it undergoes the martensite phase
transformation. Among these properties is the resistivity that decreases as the temperature of
the wire increases and hence its phase transforms to austenite. The resistance changes almost
linearly with the strain, which in turn causes a liner relationship with the angle joint rotation
q3. This resistance-angle relationship is linear because the martensite fraction is kinematically
coupled to the rotation, and the martensite fraction is what drives the resistance changes.
Further information of both SMA actuation and sensing capabilities are described in subsection
5.3.4.
5.3.2 Step 3: Fixed-wing design
The design process of BaTboT (cf. Figure 5.1) beings with the design and fabrication of a
fixed-wing that closely match the skeleton structure of the specimen. A testbed was developed
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5.3 BaTboT mechanics
1mmsilicone-membrane
half-bodyflapping-testbed
Angular acceleration VS Power
q1 deg/ s2⎡⎣ ⎤⎦
450 5 10 15 20 25 30 35 40
Pow
er [W
]
2
0
0.4
0.8
1.2
1.6
2.4
Figure 5.5: (left) half fixed-wing flapping testbed, (right) experimental quantification of power
requirements for flapping at maximum f = 10Hz. Source: The author.
aimed at testing the real requirements for flapping motion and to discover how to attach an
artificial membrane to the wing skeleton. Figure 5.5 shows the flapping testbed.
5.3.2.1 Flapping-wing mechanism
To power flapping motion, biological bats have four downstroke muscles: pectoralis major, sub-
scapularis, part of serratus anterior, and part of the deltoid. Together, these muscles constitute
about 12% of the bat’s body weight. The upstroke is powered by the deltoid, trapezius, the
rhomboids, infraspinatus, and supraspinatus (cf. Figure 2.5). These powerful muscles give
an insight about the huge power requirements for flapping flight in bats. In fact, Figure 2.3d
showed that the average power-to-mass ratio (wing mass) is about 0.04W/g, for a specimen
with a mass of 100g flapping at 10Hz. This gives a power requirement of 4W . Comparing the
biological data against the measurements carried out in Figure 5.5-right, one can appreciate
that with the half-body plus one wing (with a mass of ∼ 50g), the requirements in power are
almost the half (1.9W ) to power flapping flight at 10Hz. In Figure 5.5-right, the input power
has been calculated as (Power= I2flapR), with Iflap = 341mA the electrical current required to
power the DC motor at the required torque.
The final flapping-wing mechanism is presented in Figure 5.6. Two gears rotate at maximum
625RPM to achieve the maximum wingbeat frequency of 10Hz. Transmission bars connect
each gear to the respective shoulder joint of each wing. The axe of gears rotation have been
placed to allow each wing to cover an angle of q1 = 120o during flapping (L1 = 2.5cm).
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5.3 BaTboT mechanics
gears
Shoulder
L1
L1
q1
z1
y1
x1
rigid Transmission bars
Figure 5.6: Detailed flapping-wing mechanism. Source: The author.
5.3.2.2 Membrane issues
In this stage (fixed-wing design), a first approximation to fabricate the wing-membrane was
tackled. In biological bats, the membrane (plagiopatagium) is an extraordinary tissue. It
contains dozens of tiny embedded muscles that control the tension and even tiny hairs on
the surface to sense airflow conditions and improve on flight control. These features are hardly
matchable with available artificial skin technology (polymers, etc). In this thesis, two features of
the plagiopatagium have been selected to replicate: i) stretchable and ii) cambering properties.
Figure 5.7 details the platform for the fabrication of the artificial membrane.
The membrane should be easy-expandable and not present a high load to the SMA actuators.
In our experiments, this property has been achieved by mixing 20g of two different compounds of
liquid soft platinum silicone rubber (http://www.smooth-on.com/Silicone-Rubber-an/c2_
1115_1129/index.html), cf. Figure 5.7-left, resulting in a light and stretchy wing membrane
with a thickness of 0.1mm. The fabrication platform consists on two surfaces of aluminium
with a thickness of 2cm. Both surfaces have the same dimensions and a mass of 1Kg-each.
The goal is having heavy surfaces that facilitate the distribution of the silicone mixture. The
resultant mixture is distributed between the surfaces and using four screw-system located at
the corners is possible to compress the mixture to achieve the distribution. Also, small pieces
of aluminium of 0.1mm-thick are located between the surfaces at each corner. This allows the
mixture of silicone has a thickness of 0.1mm when compressed by the screws.
By the end of the process, it is obtained a rectangular piece of silicone-membrane with a
thickness of 0.1mm, cf. Figure 5.7-right. However, hundreds of tiny air bubbles are formed
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5.3 BaTboT mechanics
aluminium surfaces
silicone rubber
siliconemixture
screws for separating the surfaces
air-bubbles
Figure 5.7: (left) platform for wing membrane fabrication, (right) membrane issues: air bubbles
are kept within the mixture. Source: The author.
during the compression process, which drastically affect the stretchable property. Because the
silicone-membrane is very thin, any uniformity within the structure causes the membrane to
fracture. To solve this problem, a vacuum chamber has been used. Previously to distribute the
silicone mixture between the surfaces, the solution is put in the vacuum chamber for about 10
minutes. This process eliminates the air bubbles, resulting in a smooth, flat and compact piece
of silicone-membrane. Section 5.3.5 will detail this achievement and assess the performance of
the membrane in terms of stretchable and cambering properties.
5.3.3 Step 4: Articulated-wing design
Having the design of the fixed-wing (step 3), this subsection presents the design of the artic-
ulated wing. The objective is to reproduce the most influential planar joints of the bat wing
skeleton. The model/CAD of this planar joints are detailed in Figure 4.2. As previously ex-
plained, biological bats have highly articulated wings, similar to the arms and hands of humans.
This level of wing dexterity enables fascinating control over the changing shape of the wings,
having this an impact on manoeuvrability. The designed wings have six degrees of freedom, but
only two are directly actuated: flapping (q1) and morphing (q3). The rest are under-actuated
joints that rotate via metal tendons. Figure 5.8 details the articulated-wing design. All param-
eters related to wing-morphology were previously covered in Chapters 3 and 4, cf. Figure 3.7
and Table 3.5.
Figures 5.8a-b show the metal tendons that enable the motion of the metacarpophalangeal-
(MCP) digits III, IV and V, which are attached to the wrist joint. Each digit has different
radii, allowing for different rotation ranges. Using this approach, the digits open and close
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5.3 BaTboT mechanics
Elbowjoint
Carpus/wristjoint
MCP-III digitshoulder
Supraspinatus
Scapula
Teres major
triceps
Humerus
spinous
ulna
Flexor carpi ulnaris
Radius
Biceps
extensor carpi radialis
IV digit
V digit
(a) (b)
(c) (d)
metal tendons
Figure 5.8: Articulated-wing design: a-b) metal tendons are placed inside the bones for con-
necting the elbow joint with the wrist. This enables each digit rotate as a function of the elbow’s
rotation, c) cartoon of a biological contracted wing and its parts, d) ABS fabricated articulated
wing. Source: The author.
to maintain the proper tension of the wing membrane during the morphing-wing motion. To
calculate the radii that form the joint of each digit and the key proportions to maintain during
wing contraction and extension, biological data was used. Figure 5.9b shows the angles and
proportions between digits that maintain the membrane tension. The data has been extracted
from in-vivo experiments depicted in Figure 5.9a. Without the membrane attached to the
skeleton, both elbow and wrist joints can rotate free of load, cf. Figure 5.9c, however, note
from the specimen’s pictures that the wings are never completely extended or retracted, and it
is precisely the membrane that constrains these maximum values for extension and contraction.
Once the artificial membrane is attached to the robot’s wings, cf. Figure 5.9d, it causes a load
that limits the range of the morphing-wings but maintain proper membrane tension aimed at
the proper generation of lift forces.
The following sections 5.3.4 and 5.3.5 details on the morphing-wing mechanism driven by
SMA-based artificial muscles acting as biceps/triceps and also on some properties of the silicone-
based membrane.
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5.3 BaTboT mechanics
WristIII
IV
V
WingtipMCPIII
15deg
~15deg
0.032m
0.014m
0.012m0.045m
35.4deg63.3deg
y
x
reference line
III
IV
V
Brown University
(a)
(b)
(c)
(d)
Figure 5.9: a) A high camera captures the motion of each marker places along the specime’s
wings. Pictures were taking during contraction and extension of the wings, b) detailed dimensions
of wing morphology during middle downstroke. Angles and proportions are extracted from the in-
vivo experiments from (a), c) membrane-free maximum rotation of elbow and wrist joints during
wing contraction and extension, d) maximum rotation of elbow and wrist joints during wing
contraction and extension including the wing-membrane load. Source: The author.
5.3.4 Step 5: Morphing-wing mechanism
Shape Memory alloys (SMA) power the morphing-wings. As previously mentioned along this
thesis, NiTi SMA wires have been selected because the closer relation with biological muscle
fibers: they can contract upon electrical heating and recover without any thermal process
involved. SMAs are also extremely light and small, being suitable for their use as artificial
muscles that compose the whole wig structure. In addition, SMAs have sensing capabilities,
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5.3 BaTboT mechanics
tendon-like chord attaching the muscle to the elbow
3lsma 2rj
SMA_1
SMA_2-+
Isma1 Fsma-1
Fsma-2
3
Isma2
SMA antagonistic model
4 mm of stroke
38.7 mm
SMA wires attached to metal strips
Figure 5.10: Antagonistic mechanism of SMA-based muscle actuators. Source: The author.
avoiding the need for external sensors that add additional weight and decrease the payload
capacity of the robot. Figure 5.10 details the antagonistic pair of SMA Migamotor artificial
muscles.
The SMA Migamotor actuators are connected to the elbow joint, enabling a range of motion
of ∼ 60o. During the downstroke, wings are fully extended in order to maximize the area and
increase lift, whereas during the upstroke, wings are folded in order to reduce aerodynamic
drag. In our robot, this property has been mimicked by attaching the antagonistic pair of SMA
actuators to the elbow, which allow the wing to track a reference trajectory by implementing
a proper control methodology (refer to Section 6.5 in Chapter 6). In Figure 5.10, the modules
SMA1 and SMA2 represent this antagonistic configuration (artificial biceps and triceps). When
one SMA actuator contracts, the generated pull force (Fsma) is transformed into a joint torque
τ3 (elbow’s torque). The input of each muscle corresponds to an electrical current signal,
denoted as Isma1 and Isma2 respectively, which are a direct function of the input heating power
Psma(t) = Isma(t)2Rsma(t), Rsma(t) being the SMA electrical resistance.
The mechanical design shown in Figure 5.10 is crucial for the hypothesis presented in this
thesis. It is precisely the capacity of changing the wing profile which enables efficiency in flight
control. Because the relevance of the morphing-wing mechanism and its direct implication
with the control approach, Chapter 6 details experiments that assess the response of the SMA
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5.3 BaTboT mechanics
migamotor muscles and the performance of the mechanism. Some of the experiments covered
are the follows:
• Wing modulation: it presents the closed-loop architecture that achieves proper SMA
actuation via PID control, cf. section 6.1 and 6.5.2.
• SMA Migamotor actuator characterisation: experiments for the identification of
morphing-wing actuation dynamics, cf. section 6.3.
• Accuracy of SMA actuator model: it asses the response of the identified model (the
plant) VS the real response of the SMA actuators, cf. section 6.3.2.
• SMA as sensors: it presents experiments that enable the relation between SMA resis-
tance changes and SMA strain (joint motion), cf. section 6.4.
• SMA performance: it presents experimental testing of SMA response in terms of speed,
output torque and accuracy, cf. section 6.7.2.
5.3.5 Step 6: The wing-membrane
Besides the SMA morphing-wing mechanical mechanism, the wing-membrane plays a key role
during the modulation process of the wing. This section complements section 5.3.2.2 with details
regarding the testing of the silicone membrane in terms of: i) stretchable and ii) cambering
properties.
stretchable
Figure 5.11 shows empirical testing regarding the stretchable property of the artificial mem-
brane. To achieve this property, Dragon Skin Series silicones with high performance platinum
cure silicone rubbers were used. Both silicone components were mixed 1A:1B by weight or
volume and cure at room temperature with negligible shrinkage. Cured Dragon Skin is very
strong and very stretchy. It will stretch many times its original size without tearing and will
rebound to its original form without distortion. As previously highlighted, vacuum degassing is
recommended to minimize air bubbles in cured rubber. Figure 5.12 details a technical overview
of Dragon Skin silicone properties. To fabricate a thick membrane surface, it is recommended
to mix the three components of Dragon Skin, e.g. 10-medium, 20 and 30. On the other hand,
whether the objective is to fabricate a thin membrane surface, only two components are rec-
ommended, e.g e.g. 10-slow and Dragon Skin 20.
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5.3 BaTboT mechanics
(a)
(b) (c)
Figure 5.11: Testing on the stretchable property of the fabricated silicone membrane: a) ex-
tended plagiopatagium skin, embedded tiny muscles control the membrane tension, b) the fabri-
cated silicone membrane is attached to the wing skeleton using Sil-Poxy RTV, c) the fabrication
process described in section 5.3.2.2 results on a light artificial skin with the required stretchable
property. Also note that after vacuum, air-bubbles are eliminated. Source: The author.
For preparation, it is recommended to use a properly ventilated area (room size ventilation).
Wear safety glasses, long sleeves and rubber gloves to minimize contamination risk. Wear vinyl
gloves only. Latex gloves will inhibit the cure of the rubber. Store and use material at room
temperature (73F/23C). Warmer temperatures will drastically reduce working time and cure
time. Storing material at warmer temperatures will also reduce the usable shelf life of unused
material. These products have a limited shelf life and should be used as soon as possible. Before
mixing the components, pre-mix one thoroughly. After dispensing required amounts of parts
A and B into mixing container (1A:1B by volume or weight), mix thoroughly for 3 minutes
making sure to scrape the sides and bottom of the mixing container several times. After mixing
parts A and B, vacuum degassing is recommended to eliminate any entrapped air. Vacuum
material for 2-3 minutes (29 inches of mercury), making sure that you leave enough room in
container for product volume expansion. It is not recommended to exceed 10% by weight of
total system (A+B).
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5.3 BaTboT mechanics
Figure 5.12: Technical overview of Dragon Skin silicone properties. Source: http://www.
smooth-on.com.
Figure 5.13: Mid-downstroke wing camber and angle of attack are estimated as follows: (A) A
parasagittal (xg-zg) cross section of the wing was taken at the yg-value of the wrist at the time
of maximum wingspan. Six triangular sections of the wing membrane crossed that plane and the
intersections of triangle borders in the plane (red circles) were used as estimates of membrane
position. (B) The actual curved shape of the membrane in the plane (solid black line) was
estimated using the first term of a sine series fitted to those seven points. The maximum distance
of the membrane line from the chord line (dashed grey line) was divided by the length of the chord
line to give wing camber. (C) Angle of attack (α) was calculated as α1+α2, where α1 is the angle
of the wing chord line above horizontal (blue dashed line), and α2 is the angle between horizontal
and the velocity vector of the wrist (red arrow) in the xg − zg plane. Source: (20).
Cambering
In (20), wing camber has been quantified for several species of biological bats. The results of
the study have shown the influence of bat body mass (mt) not only on the wing camber, but
also on the lift production, flight velocity, and flapping frequency. Figure 5J in (20) shows the
wing camber coefficient as a function of body mass. For the specie Rousettus aegyptiacus, wing
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5.3 BaTboT mechanics
camber at maximum span is proportional to ∝ m0.71t . Figure 5.13 has been extracted from
(20) and explains the procedure to measure the mid-downstroke wing camber using an external
camera. Following the relationm0.71t , beingmt = 0.079kg the mass of BaTboT without battery,
wing camber has been empirically adjusted to be about 0.16, similar to the values measured
from the biological counterpart (0.14 − 0.15). Figure 5.14 shows wing camber values. The
variation of the maximum camber of the membrane airfoil shape with respect to time is shown
for a representative case (Vair = 4.1ms−1). Over a significant portion of the wingbeat cycle
(arguably the portion during which most of the aerodynamic forces are generated by the wing),
the camber is relatively constant and hovers around the value of 10% of the chord length. The
spikes in the value of the camber near the top of the upstroke are to be expected since wing
will fold significantly during this portion of the wingbeat.
Figure 5.14: Variation of camber during the wingbeat cycle. Source: (25).
On the other hand, we have performed aerodynamics measurements in the wind-tunnel for
testing how the membrane behaves when subject to different airflow speeds. These experimental
results are reported in Figure 5.15. The robot is mounted in the wind tunnel, on the end of
a supporting sting that defines the angle of attack. The robot is mounted on top of a 6-DoF
force sensor from which both lift CL and drag CD coefficients are experimentally calculated as
a function of the airflow speeds and angle of attack. Typical results are shown in Figure 5.15.
In Figure 5.15a the membrane is fixed to a support base and camber is not adjusted. Because
of the airflow (Vair = 5ms−1) at an angle of attack varying from 10− 20o, an excessive camber
value is measured with the camera (0.48). Lift and drag coefficients (CL, CD) are measured
between the range of interest (angle of attack varying from 7 − 13o). On the other hand, in
Figure 5.15b, the same membrane surface is attached to BaTboT wing’s skeleton and camber
is adjusted to be about 0.16. Note the robot is not flapping, the wings are fixed to middle
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5.4 BaTboT electronics and sensors
(b)
Force sensor
airflow
FL
FD
excesivecambering
load cell
CL
CD
airflow propercambering
(a)
(b)
CL
CD
Figure 5.15: Implications of wing camber into lift and drag production (Vair = 5ms−1, fixed-
wing): a) excessive wing camber (0.48), b) proper wing camber (0.16). Source: The author.
downtroke. Lift and drag coefficients are again measured under the same airspeed conditions
and note there has been an increment of 65% in lift coefficient value (at angle of attack of 9o).
Further experiments in Chapter 6, cf. Figure 6.4 shows quantification of the membrane
loads caused during the extension process of the wings.
5.4 BaTboT electronics and sensors
BaTboT is composed by a simple electronic configuration mainly because each added com-
ponent represents an additional weight to the overall structure. The challenge is to design an
onboard hardware architecture light and power enough to enable the proper control of the robot
towards its autonomous operation. This section details the onboard hardware architecture, its
components, functions, and the power consumption of the robot.
5.4.1 Arduino controller-board
Arduino is one of the most extended, simple and robust commercial solutions of micro-controllers.
The ArduinoNano version (http://arduino.cc/en/Main/ArduinoBoardNano) is one of the
lightest chips powered by an Atmel ATmega168 running at 16MHz. It has an operation volt-
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5.4 BaTboT electronics and sensors
+
Figure 5.16: Micro-controller and Migamotor SMA muscle connection diagram. Source: The
author.
age of 5V , 14 I/O digital pins (6 are PWM) and 8 analog pins. The ATmega168 has 16KB of
flash memory for storing code (of which 2KB is used for the bootloader). Arduino provides a
software (http://arduino.cc/en/Main/Software) for programming the micro-controller. An-
nex 10.5 details the control-code and Kalman filter code programmed into the Arduino Nano
for driving BaTboT behaviour, whereas Figure 5.16 shows the connection diagram between the
micro-controller and the Migamotor SMA actuator.
The SMA actuator can provide simple full-range motion by simply having their CNTRL pin
enabled. The Migamotor SMA muscle will keep contracting/rotating as long as the CNTRL
pin is held high until it reaches its full extent. The rate at which this motion will occur will vary
depending on, the load, friction in any linkage, voltage level and ambient temperature. The
recommended way of controlling speed is using a Pulse Width Modulation (PWM) signal to the
CNTRL pin rather than simply setting it. The speed of motion will be governed by the duty
cycle of this PWM signal. The larger the duty cycle, the faster the Migamotor SMA muscle will
move. Within certain constraints the actual period of the PWM does not have any affect. Unlike
the control of some other devices, Migamotor SMA muscle actuator can operate at relatively
low periods (typically to between 100Hz and 1KHz). The Migamotor SMA muscle actuator
has a position feedback pin that gives an analog signal that can be interpreted to provide an
approximate position. The change that must be measured is the change in resistance between
the position feedback pin and the Vsma pin of the Migamotor SMA muscle. This resistance will
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5.4 BaTboT electronics and sensors
vary around 10% during the contraction of the Migamotor SMA muscle. There are a number
of methods that can be used to measure this resistance. A typical circuit, cf. Figure 5.16 used
to measure this resistance is to time a capacitor charge up that is inserted in series with the
resistance of the position control pin.
In Figure 5.16, setting the CNTRL signal high of the Migamotor SMA muscle will drive the
position feedback pin low and cause the capacitor to discharge. Clearing the CNTRL signal
will start the capacitor charging. In order to be able to accurately time this charge up period,
the circuit uses a comparator with a Voltage Reference input that will trigger an interrupt in a
micro-controller when the reference voltage is reached. The code required for a resistance based
position control system configured around this is very similar to that of the optical encoder.
The only major difference is that instead of reading pulses from the encoder in an interrupt
handler, the feedback resistance using the capacitor circuit must be periodically measured. The
pseudo code for such a reading is detailed as follows:
1. Set Voltage Reference to low value (e.g. 0.8v)
2. Set Migamotor SMA muscle CNTRL Pin High to discharge capacitor
3. Wait for Comparator to indicate Voltage has dropped below VRef
4. Set Voltage Reference to mid value (e.g. 1.5v)
5. Set Migamotor SMA muscle CNTRL Pin Low to start capacitor charging Wait for Comparator to indicate
Voltage has risen above VRef Start Timer
6. Set Voltage Reference to high value (e.g. 3.5v)
7. Wait for Comparator to indicate Voltage has risen above VRef Stop Timer
8. Resistance = Timer Value
By measuring the SMA resistance changes is possible to estimate the position state driven
by the actuator and therefore to close the control loop. This SMA Resistance-to-Motion rela-
tionship (RM) is introduced in Chapter 6 section 6.4, whereas the control algorithm is detailed
in section 6.5.4.
5.4.2 The Inertial Measurement Unit (IMU)
The Inertial Measurement Unit is a digital combo board of 6 Degrees of Freedom with gyros:
ITG3200 and accelerometers: ADXL345 (https://www.sparkfun.com/products/10121?). IMU
readings are vital for the feedback of attitude variables that enable roll and pitch control of
BaTboT. Details about the use of the IMU merged with the control architecture can be found
in Chapter 6, Figure 6.14. This chip is ready to be connected to the Arduino board, the
unique requirement is to filter the IMU data for reducing noise. This procedure is shown in
113
5.4 BaTboT electronics and sensors
roll IMUroll IMU filtered
pitch IMUpitch IMU filtered
sample
attit
ude
[deg
]
Figure 5.17: Comparison of attitude IMU readings before and after Kalman filtering. Source:
Arduino.
Figure 5.17. Kalman filtering technique is used for both reducing noise and predicting attitude
motions. Annex 10.5 details the code.
5.4.3 SMA power drivers
The Miga Analog Driver V5 (MAD-V5) is a MOSFET switch designed to safely power the Mig-
amotor SMA actuators across a wide range of speeds or input voltages. This driver generates
the current signal based on the digital control command sent from the Arduino board. The
schematic of the circuit is described in Figure 5.18. The MAD-V5 allows either push-button
operation, or external GATE (CNTL) signals to actuate the Migamotor SMA actuator until
the END limit is reached (goes LOW). The MAD-V5 then cuts power momentarily, preventing
overheating of the SMA wires. The Gate transistor allows up to 30V input, but it is rec-
ommended to use logic (2.5 to 5-Volt) levels. Pulse-Width-Modulated (PWM) signals can be
applied at the Gate to control the actuation speed for a set voltage, or even an AC driven
current signal mounted on a DC level. For instance, the application of +28VDC power to the
Migamotor SMA actuator would result in very fast actuation (∼ 70ms). In Figure 5.18, JP1 is
the power supply and/or micro-controller connector, JP2 is connected to the Migamotor SMA
actuator. Maximum peak current: 7A, maximum continuous current: 5A.
114
5.5 BaTboT consumption
Control signal
SMA-mosfet power
1.5cm
Figure 5.18: Miga analog driver V5 pinout diagram. Source: The author.
5.5 BaTboT consumption
Consumption is measured in terms of required electrical currents. Figure 5.19 shows the per-
centage of current consumption of each electronics component of BaTboT, whereas Table 5.5
consigns the numerical values. As expected, the SMA actuators require most of the input
current, about 78.7%.
Table 5.5: General values of current consumption
Component Required current [A] % of consumption
SMA actuators (x4) 0.6x4 78.7%
DC motor (x1) 0.35 13%
Arduino board (x1) 0.05 3%
IMU (x1) 0.1 5%
Total consumption = 2.9
115
5.6 BaTboT costs
Figure 5.19: Percentage of current consumption per component. Source: The author.
5.6 BaTboT costs
Table 5.6 details the costs of the components that are involved in the fabrication process of the
robot.
Table 5.6: Fabrication costs
Component Manufacturer/seller Item cost Quantity used Total
SMA Nanomuscle NM70R-6P Migamotor 6.93e 4 27.72e
Miga Analog Driver V5 (MOSFET) Migamotor 10.75e 4 43e
DC adjustable conversor (4-25V-2A) BricoGeek 10.40e 1 10.40e
50:1 Micro Metal Gearmotor HP Pololu 12.30e 1 12.30e
IMU 6DOF (ITG3200/ADXL345) Sparkfun 50e 1 50e
ArduinoNano V3.0 Arduino 17.95e 1 17.95
Polymer Lithium Ion Battery - 850mAh Sparkfun 6.90e 1 6.90e
(ABS) plastic material - - - 60e
Aluminium plates for membrane molde alu-stock 40e 2 80e
Steel wire rope for metal tendons mcmaster 5e 1 5e
Sil-Poxy Silicone Adhesive smooth-on 22.13e 1 22.13e
Dragon Skin Silicone Rubber for wing membrane smooth-on 23.20e 2 46.40e
Total costs =381.8e
5.7 Remarks
This Chapter has completely described the bio-inspired design-flow applied to the development
and fabrication of BaTboT. The criteria for design have been classified in terms of morphology,
kinematics, dynamics and aerodynamics parameters extracted from the analysis of biological
data. Most importantly, the data has been related as a function of design parameters such as
overall mass, wingspan, body mass, wing mass, wing lenghts, etc. These relations have enabled
116
5.7 Remarks
the definition of a framework that can be applied to future designs of BaTboT. The Chapter has
done an special emphasis into the design of the highly articulated morphing-wings, showing how
to mechanically achieve the wing modulation required for testing the hypothesis proposed at
the beginning of this thesis. The following Chapter introduces the characterisation of the main
components introduced herein and the control methods that are key for the proper regulation
of the wing profile. All of this towards future developments that achieve autonomous flight.
117
6
BaTboT Control
6.1 Control goal
The control goal is twofold: firstly, enabling BaTboT of accurate forward and turning flight
capacity, and secondly achieving efficient flight by properly driving wing modulation. First of
all, the control scheme must ensure the robot is able to track wing joint trajectories defined in
Figure 4.5 (forward flight), and Figure 4.6 (turning flight). In case of forward flight maneuver,
when both wings are flapping symmetrically and keeping the same shape-configuration, the
resultant torques produced at the robot Center of Mass (CM) are theoretically cancelled between
each other. In the practice though, a slight mechanical bias caused within the fabrication
process introduces asymmetries that only could be regulated by means of feeding back attitude
measurements (roll and pitch) and adjusting the attitude tracking error. In case of turning
flight maneuver, wings flap in different shape-configurations aimed at displacing the CM and
therefore causing the attitude variation of the body. The control must monitor the center of
mass is on the desired place.
This chapter introduces the approaches-to-follow to fulfill both goals. To this purpose a
Flight Control Architecture (FCA) is formulated in Figure 6.1. It is composed by two control
strategies:
1. Morphing-wing controller: it consists on an inner closed-loop position control which drives
the changes on wing configuration via Shape Memory Alloy actuation. Wings are mod-
ulated depending on the wing joint references trajectories and an outer attitude control
loop commands. The control goal is to efficiently drive Shape Memory Alloy ac-
tuators aimed at the fastest modulation of the wing shape, i.e, contraction and
118
6.2 Flight Control Architecture (FCA)
SMAactuation
Rsma=Vsma/Isma
SMA Resistancemeasurement
Attitude control
Motors
-+ Morphing
control
outer loop
q3
R/C (flapping)IMU
Bat robot
+-
R/C (attitude)forward/turning
inner loop
[Uheating]
[q1…q6]R,L[roll, pitch]
q3 = f (R)
q1,ref
φ,θ[ ]
uφ ,uθ⎡⎣ ⎤⎦
wristmapping
Elbow motionestimation
uθ = q2,ref
uφ = q3,ref
Wing trajectories
τ 3
τ1τ 2
τ 3τ 4τ 5τ 6
Figure 6.1: Flight Control Architecture (FCA). Source: The author.
extension. Efficiency is evaluated as a tradeoff between SMA output torque
and power consumption that achieve the fastest SMA actuation speed.
2. Attitude controller: It drives the inner control loop. It consists on an outer closed-loop
position scheme based on a novel backstepping+DAF controller. It regulates the roll
(φ) and pitch (θ) response of the robot and allows forward and turning maneuvers. The
control goal is to consider the influence of wing inertia into the attitude control
law aimed at efficiently driving the wing modulation. Efficiency is evaluated
in terms of thrust production.
6.2 Flight Control Architecture (FCA)
Inner loop: morphing-wing modulation
It is composed by six modules:
• Motors: it refers to the DC-motor models that power the primary flapping motion q1
and the rotation of wings forward/backward about the gravity axis q2. The latter allows
for the pitch motion of the body θ. Outputs are torque commands to the inertial model;
τ1, τ2.
• SMA actuation: it refers to the Shape Memory Alloy antagonistic model. SMA actuators
are driven by the control signal uheating which corresponds to the input heating power.
119
6.2 Flight Control Architecture (FCA)
Each SMA contracts upon electrical heating generating a pulling force on the elbow joint
of the wing. The output is the joint torque that rotate the elbow joint: τ3.
• Wrist mapping : it maps joint torques τ4, τ5, τ6 from elbow torque τ3. This is due to wrist
jointsq4, q5, q6 are under-actuated. The mapping is based on the kinematics configuration
of the wrist joint shown in Figure 3.7b.
• Morphing-wing : it refers to the control strategy that drives SMA actuators. In this
chapter two control techniques are evaluated: i) a sliding-model nonlinear control and ii)
a PID linear control. Inputs are the position errors of elbow joint motions of both wings
(q3,ref − q3)R,L and outputs are the control signals uheating that drive SMA actuation.
• Elbow motion estimation: it refers to the Resistance-Motion relationship between SMA
electrical resistance Rsma and joint motion q3. Besides actuators, SMAs can also be used
as sensors. It turns out that Rsma changes lineally with respect to the strain rate of
the SMA wire. Based on this, elbow motion can be estimated as a function of electrical
resistance measurements: q3 = f(Rsma).
• SMA resistance measurements : it refers to the calculation of SMA electrical resistance
Rsma by measuring the applied voltage and electrical current through the SMA wires.
The inner loop runs at 30Hz. It allows a maximum SMA physical actuation of 2.5Hz
mainly due to the limitations of SMA actuation caused by slow cooling time of the wires. This
upper limit in actuation speed was previously estimated by simulations from Figure 4.22a, cf.
Table 4.9. Experiments carried out in section 6.3 confirm this limit value using the real system.
Further details of the inner loop control can be found in section 6.5.
Outer loop: attitude regulation
It is composed by three modules:
• Wing trajectories : it refers to joint trajectory generator module. It generates kinematics
profiles q1...q6 for each wing depending on the maneuver type: forward or turning flight.
Profiles are those previously described in Figures 4.5 and 4.6 respectively.
• Attitude control : it refers to the control strategy that drives the attitude regulation (φ, θ)
and manages the inner loop. Inputs are the desired joint trajectories and attitude ref-
erences. Outputs are control signals that drive the inner loop, allowing for proper wing
modulation. The attitude control is based on a novel backstepping+DAF methodology,
120
6.2 Flight Control Architecture (FCA)
which uses a Desired Angular Acceleration Function (DAF) to incorporate the influence
of wing inertia into the control law.
• IMU : it refers to the Inertial Measurement Unit sensor, which allows feeding back attitude
variations. In the practice, the IMU sensor includes 3-axis accelerometer and 3-axis
gyroscopes1.
The outer backstepping+DAF attitude control runs at 20Hz and generates proper control
references that allow the wing modulation.
The methods for control
1. Experiments on frequency response analysis are carried out for the identification of the
SMA actuation module, cf. Figure 6.1. System identification is aimed at characterizing
the relationship between applied power input (via electrical current heating) and SMA
output torque. Section 6.3 details the experimental methods.
2. Experiments are carried out to characterize the Resistance-Motion (RM) relationship
between SMA electrical resistance changes (Rsma) and SMA strain. This allows the
identification of the Elbow motion estimation module in Figure 6.1. The RM relationship
makes possible to feedback elbow motion q3 and close the inner loop. Section 6.4 details
the methods.
3. The Morphing wing control module is formulated (inner loop). Two control methodologies
are first tested on simulation, one based on sliding-mode technique and the other based
on classic PID control. Also, mechanisms that enhance the performance of the SMA
actuators are complemented into the morphing controller. These mechanisms are called
anti-slack and anti-overload. Section 6.5 details the methods.
4. The Attitude control module is formulated (outer loop). It shows how the properly reg-
ulate roll and pitch variables (φ, θ) based on wing modulation. A novel strategy called
backstepping+DAF is introduced. Using the inertial model from Eq. (4.14) and (4.15), a
Desired Angular Acceleration Function (DAF) is formulated. The DAF uses the inertial
model to incorporate wing inertia information that is key for the generation of body accel-
erations. These accelerations are used as references to the backstopping+DAF controller
1The IMU-model placed on the robot corresponds to ITG3200/ADXL345, http://www.sparkfun.com/
products/10121.
121
6.3 SMA actuation: experimental characterization
aimed at improving the modulation patterns of wing kinematics. Section 6.6 details this
novel control.
6.3 SMA actuation: experimental characterization
6.3.1 Frequency analysis
Previously in Chapter 4 section 4.5.2 a phenomenological model was used to describe the com-
plex nature of SMAs, especially in terms of their thermomechanical behaviour and the hys-
teretic effects. This becomes useful for analyzing over-heating effects and other aspects such
as over-stressing, that could be impractical through real experimentation. However, to control
purposes, phenomenological models are mostly useless. The main reason concerns that some
of the internal state variables of the SMAs are irrelevant to control, such as temperature or
martensite ratio. Actual measurements of these variables are often difficult, even impractical.
On top of this, phenomenological models describe the large-signal behavior of SMAs, which is
quite nonlinear, hysteretic and often not repeatable.
To overcome these problems, remarkable investigations in (9) and (67) about the frequency
response analysis of NiTi SMA actuators were carried out aimed at the development of linear
force models that attempted to characterize the relationship between applied power and SMA
output force. This relationship is actually more suitable than phenomenological models for the
development of controllers that drive SMA actuation based on electrical heating. The procedure
described in (9) demonstrated that the AC frequency response of SMA NiTi wires is similar
to that of a first-order linear system.This leads to the analyzes of SMA small-signal behavior,
rather than the large-signal response.
This section briefly describes the methods for obtaining a SMA force model relating output
force -to- input heating power. As demonstrated in (9), the frequency response analysis allows
for the study of the small-signal response of SMA wires over a suitable frequency range, which
has been observed to be very repeatable and also exhibits very little hysteresis. So, the the goal
here consists on finding the linear model plant that matches the behavior of the Migamotor’s
SMA nano-muscles in terms of input power to output torque. Frequency response analysis is
carried out by applying sinusoids test signals to the input acting as heating power, and looking
at the output to see how the system responds by measuring the output force. Therefore, the
model to be identified can be represented by a transfer function that matches the measured
gain and phase data over the frequency range of interest. Here, the input signal command
122
6.3 SMA actuation: experimental characterization
IsmaDAC
16-bit ADC + anti-aliasing filter9620-05-ATI
Fixed wing testbed
Low passfilter
Torqueconversion
Fτ3
Currentconversion
uheating Transconductanceamplifier
lr
q3~60º
F
Force sensor
τ3
SMAs
Figure 6.2: Experimental testbed for the characterization of SMA input power (uheating) to out-
put torque (τ3). Forces (F ) are measured using a force sensor with 0.318 gram-force of resolution.
Torque conversion is applied by considering the humerus bone length (lr), as: τ3 = Flr[Nm]. This
allows the identification of SMA actuation. Source: The author.
represents the heating power, which can be described as: uheating = a+ bsin(ωt). The term a
is the mean input power, b is the magnitude of the sinusoidal component, and ω = 2πf , being
f the commanded frequency for SMA contraction.
Experimental Testbed
Figure 6.2 details the experimental setup for the identification of the SMA actuation. Applied
heating power uheating is converted to a current signal using a nominal value for the SMA wires
electrical resistance provided by the manufacturer Rsma = 8.5Ω (cf. www.migamotors.com),
and is then converted to an analog voltage and sent to the precision transconductance amplifier
that drives the SMA actuator. The force response is measured using a Nano 17 transducer
ATI Industrial Automation force sensor with0.318 gram-force of resolution. In addition, a
16-bit DAC with embedded 1st order anti-aliasing filter with a bandwidth from 5KHz to
10KHz is also used. Details about the specifications of the force sensor can be found at:
http://www.ati-ia.com/products/ft/ft_models.aspx?id=Nano17.
The force signal F is recorded into the computer, and then mapped to a torque value τ3.
This corresponds to the elbow torque. Note the force measurements have been carried out
taking into consideration the payload of the bones and the friction of wing joints1, but the
membrane load has not been taken into account. Later in section 6.3.2 it will be demonstrated
that thanks to the highly stretchable and light properties of the membrane, the resultant data
from the identification process is not significantly affected.
1Joint friction is not directly measured or quantified.
123
6.3 SMA actuation: experimental characterization
The identification procedure
To identify the system, the magnitude and phase of the sine-wave component of F that is
generated upon heating driven by uheating = a+bsin(ωt) is measured over a frequency range of
0.1 to 100Hz. It is important to highlight that the DC component of uheating (a) should be keep
constant over the frequency range. To extract the sine-wave component in the recorded force
data F a least-squares methodology is used. The least-squares-based parameter identification
algorithm is an extended methodology widely applied in several systems. A detailed step-by-
step use of this method can be found for a similar SMA application in (9). Here, a briefly
introduction of the least-squares methodology is presented. Considering the measured force
data F (t) of the SMA is of the form:
F (t) = A cos (ωt+ ϕ)F (t) = A [cos (ϕ) cos (ωt)− sin (ϕ) sin (ωt)] ,
(6.1)
only two variables are known: i) the measured force data F (t), which is produced when the
SMAs are pulling the elbow joint when contract upon heating (uheating), and ii) the driving
frequency ω = 2πf , being f the actuation frequency of uheating. However, the force amplitude
A, and the phase of the signal ϕ are unknown. The goal is to estimate both parameters.
In order to put together the unknown parameters, a parametric model is used in (6.2), being
z (t) = F (t). Note both unknown parameters A, ϕ are now contained into vector θ∗.
z (t) = θ∗Tφ(t)z (t) = [A cos (ϕ) , A sin (ϕ)]
T[cos (ωt) ,− sin (ωt)]
T (6.2)
By using the Adaptive Control Toolbox in ©Matlab, a least-squares algorithm has been
used to estimate: θT = G(t)e(t), where G(t) is a gain vector and e(t) corresponds to the
estimation error: e(t) = z(t) − z (t). The term z (t) corresponds to the parametric estimation
model z (t) = θTφ(t). In other words, the error reflects the distance between the estimate θ
and the unknown θ∗. The least-squares algorithm fits a mathematical model to a sequence of
observed data by minimizing the sum of the squares of the difference between the observed and
the computed data. Once θ∗ has been estimated as θ = [A1, A2]T , being A1 = Acos(ϕ), and
A2 = Asin(ϕ), the estimation of A and ϕ of force amplitude A and phase ϕ is defined as:
A =√A2
1 + A22,
ϕ = tan−1(A2
A1
) (6.3)
124
6.3 SMA actuation: experimental characterization
Mag
nitud
e (d
B)Ph
ase
(deg
)
Bode Diagram
Frequency (rad/s)
Experimental
Frequency [rad/s]
Experimental
Model
Model
Phas
e [d
eg]
Mag
nitu
de [d
B]
Figure 6.3: (Experimental VS model) Bode magnitude and phase plots for NiTi 150μm SMA
Migamotor actuators. The insets show several experimental measurements of magnitude and
phase. Magnitude is given by: 20log(A/b), where A is the least-squares estimation of the force
amplitude measured using Eq. 6.3, and b is the AC power of the input signal uheating. Phase is
given by the term ϕ in Eq. 6.3. The transfer function that fits the experimental data is shown
in Eq. 6.4. This plot also compares the model in Eq. 6.4 against the experimental data. Source:
The author.
After running the identification algorithm, Figure 6.3 shows the experimental Bode plots.
The magnitude of the curves is given by 20log(A/b), where A is the least-squares estimation of
the force amplitude, and b is the AC power of the input signal Uheating. Note the SMA frequency
response in Figure 6.3 is similar to that of a first-order linear system, where the frequency at
which the change of slope in magnitude occurs is known as a pole (s = −2.857). Several
measurements have indicated that the suitable transfer function that fits the experimental data
is:
τ3(s) = 0.016(0.35s+ 1)−1uheating(s) (6.4)
By using the model in Eq. (6.4), the AC behavior of SMA actuation has been experimentally
identified in the form of a Laplace transfer function that relates the input heating power uheating
125
6.3 SMA actuation: experimental characterization
with the SMA output torque generated at the elbow joint τ3. The following section explores
the accuracy of the identified SMA actuation model to control purposes.
6.3.2 Experimental validation of SMA actuation model
Using the setup from Figure 6.2, this section introduces a comprehensive experimental review
on the accuracy of the identified SMA actuation model in Eq. (6.4). Three experiments are
carried out:
1. The membrane-load : The fabricated silicon-based membrane produces an external load to
the SMA actuator. Despite this load is minimum due to the light and stretchable property
of the membrane material, it seems convenient to quantify and verify the capacity of the
SMA actuators interns of force production regarding the membrane load. Quantification
is achieved by measurements of output torque (τ3) and required input power (uheating).
2. Morphing-wing frequency : SMA actuators allow the wings to contract and extend via
elbow and wrist rotations. This is known as morphing-wings. Here, nominal and over-
loaded operation modes of the SMA actuators are evaluated in terms of actuation speed.
Actuation speed is experimentally calculated by measuring the rotation speed of the el-
bow joint (θ3). Also, experimental data is compared against simulations with the SMA
phenomenological model (cf. Figure 4.22a and Table 4.9).
3. SMA actuation model accuracy : It compares the nominal and overloaded response of the
model in Eq. (6.4) against experiments of small-signal response of the SMA actuators at
driven frequency of f = 2Hz.
The membrane-load
Figure 6.4 shows experimental measurements of generated SMA output torque at elbow joint
(τ3) in response of nominal and overloaded values of input heating power (uheating). The
applied heating power corresponds to electrical current bias values ranging from Isma = 300mA
(nominal) to Isma = 550mA (overloaded). Theoretically, Figure 4.22a in Chapter 4 defined that
in order to achieve the required rotation speed of the elbow joint (θ3) under both nominal and
overloaded modes of SMA operation, DC input currents ranging from I = 350mA to I = 550mA
should be applied. Simulations indicated the elbow joint is able to rotate ∼ 60o in 200ms
when applying a nominal power of uheating =∼ 1.04W , and by increasing the applied power
to uheating =∼ 2.57W :∼ 60o in 100ms (overloaded). In the practice thought, joint frictions
126
6.3 SMA actuation: experimental characterization
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.0180
0.5
1
1.5
2
2.5
3
3 [Nm]
Powe
r[W]
Measurement Measurement
u heatingW [
]
τ3τ3
Figure 6.4: (Above) right wing extended and contracted taking into account the load produced
by the silicon membrane. (Below) Experimental measurements of SMA output torque τ3 generated
by input heating power uheating. Values are classified by nominal and overloaded operation mode
of the SMA actuators. Nominal behavior is achieved by applying uheating =∼ 1.36W whereas
overloaded behavior requires an input power of uheating =∼ 2.57W . Inset plots show the average
peak of produced elbow torque at both SMA operation modes. Source: The author.
and the anisotropic loading of the membrane make the input power to increase with respect
to the simulation predictions. In the experiments, under nominal mode (uheating =∼ 1.36W )
the output torque τ3 stabilizes around ∼ 0.007 − 0.008Nm whereas under overloaded mode
(uheating = 3.06W ) around ∼ 0.016−0.02Nm. It has been observed in Figure 6.6 about 20% of
increase in the input heating power to produce the desired output torque that achieves elbow
joint rotation speeds shown in Figure 6.5. It is precisely the high stretchable property of the
membrane material that allows for minimum load. The following section emphases in this issue.
Also, numerical values comparing simulation and experimental data can be found in Table 6.1.
Morphing-wing frequency
Figure 6.5 presents the average results of several elbow joint rotation speeds (θ3) achieved af-
ter applying both nominal and overloaded profiles of input power uheating. Root mean square
127
6.3 SMA actuation: experimental characterization
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Measurement
3[deg
/ms]
OverloadedNominal
0 1 2 3 4 5 6Measurements
0
q 3deg/ms
[]
0.10.20.30.40.50.60.7
0
Figure 6.5: Experimental measurements of joint rotation speeds of elbow joint (θ3) that are
obtained by applying heating power values at nominal (uheating = 1.36W ), and overloaded
(uheating = 3.06W ) SMA operation. Source: The author.
(RMS) joint-speed values have been measured using the -contracted - and -extended - pins pro-
vided by the SMA artificial muscles (cf. section 5.3.4). These pins activate when the actuator
reaches the maximum stroke (contracted) and when it returns to the neutral position (ex-
tended). In the nominal mode of SMA operation, the joint speed average is ∼ 60o in 225ms
(0.26deg/ms), whereas in the overloaded mode, ∼ 60o in 120ms (0.5deg/ms). Thereby, regu-
lating SMA operation within the overloaded region enables the wings to contract and extend
at a maximum frequency of f = 2.5Hz. This is known as the morphing-wing frequency. This
value is obtained as follows: (tdown + tup + tcool = 400ms)−1, where tdown and tup are the
times involve in the downstroke and upstroke cycles of the wing (max. speed of each cycle
set at 120ms) and tcool = 160ms is the death time required to enable the active SMA wire to
recover its original length during the cooling process. The death time is important within the
antagonistic configuration of SMA actuators since it contributes avoiding lock issues. This will
be explore later in section 6.5.
SMA actuation model accuracy
The nonlinear SMA phenomenological model described in Chapter 4 has been useful to ad-
dress some important limitations of SMA thermo-mechanics, at least in terms of simulation.
Previously in Eq. (6.4), a frequency response analysis allowed for the identification of the AC
behavior of the SMA actuators. To verify the accuracy of the model upon small-signal heating
128
6.3 SMA actuation: experimental characterization
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
3 [Nm]
t[s]
uheating=3.06W(overloaded)
uheating=1.36W(nominal)
ExperimentModel
Figure 6.6: (Experimental) Input Power to Output Torque small-signal response of the SMA
actuators, being uheating = a+ bsin(2πft), f = 2Hz. Source: The author.
power, Figure 6.6 compares the time-response of the model in Eq. (6.4) against experimental
data measured using the setup in Figure 6.2. By measuring the output torque generated by the
power signal of the form: uheating = a + bsin(2πft), it is possible to compare how the model
behaves for both nominal and operation modes of the SMAs. Nominal operation is when the
DC bias of input power is a = 1.36W whereas overloaded mode is when a = 3.06W . The small-
signal amplitude is defined by the parameter b which in both cases corresponds to b = 2mW .
The driven frequency of the results shown in Figure 6.6 corresponds to f = 2Hz. Similar results
have been observed with other ranges of driven frequencies.
6.3.3 Data summary
Table 6.1 summarizes the average numerical values of SMA actuation performance in terms of:
i) actuation speed, ii) input heating power, and iii) output torque.
The maximum allowed input current to drive SMA motion is set to Imax = 550 − 600mA.
This limitation is defined in order to avoid the SMAs to over-heat and therefore to over-stress.
In Table 6.1, simulation data indicates that the resultant nominal joint speed of 0.3deg/ms
results from applying an input heating current bias of I = 350mA (input power of 1.04W ).
During the experiments a slightly increase in power is required to maintain a similar joint
speed (0.26deg/ms). The difference in power consumption is due to non-modeled joint friction
and also from non-modeled membrane-load effects during wings extension. In spite of this, the
129
6.4 SMA Resistance-to-Motion relationship (RM)
Table 6.1: Summary of SMA actuation performance.
Parameter Theoretical1 Simulation2 Experimental3
Nominal
Joint speed q3 [deg/ms] 60/300 60/200 ∼ 60/225
Input heating power uheating [W ] 0.26 1.04 0.87 - 1.5
Output torque τ3 [Nm] 0.0012 0.0007 0.008
Overloaded
Joint speed q3[deg/ms] —- 60/100 ∼ 60/120
Input heating power uheating [W ] —- 2.57 2 - 3.06
Output torque τ3 [Nm] —- 0.0027 0.020
1 Nominal values provided by Migamotor’s model NanoMuscle RS-70-CE 1131 (96).2 Simulation results from Figure 4.22a (joint speed and input heating power), and Table 4.5 (output torque)3 Experimental results from Figures 6.5 (joint speed) and 6.6 (input power and output torque).
0 10 20 30 40 50 601
2
3
4
5
6
7
8
9
R [
Oh
ms]
θ3 [deg]
RM, To=23oC
RM, To=22.83oC
RM, To=22.75oC
RM, To=22.7oC
RM, To=22.6oC
R sma
Ω []
q3 deg[ ]
SMAs
Vair
Figure 6.7: (Experimental) Resistance-Motion (RM) linear relationship between SMA electri-
cal resistance change (Rsma) and the angular motion generated at the elbow joint (q3). Small
variations in ambient temperature (To) modify the RM relationship. The inset shows BaTboT
in the wind-tunnel. Ambient temperature has been measured with a MS 1000-CS-WC tempera-
ture sensor supplied by ATS (http://www.qats.com/Products). Changes in Rsma are constantly
measured during the experiment. Source: the author.
important issue concerns to keeping the input current below 600mA. Similar results have been
obtained during overloaded operation. Limitations in input power should be driven by a proper
SMA controller. Section 6.5 will present the control approach.
130
6.4 SMA Resistance-to-Motion relationship (RM)
6.4 SMA Resistance-to-Motion relationship (RM)
Although SMAs are mostly used for actuation, they also have sensing capabilities. Several
properties of the SMAs change as it undergoes the martensite phase transformation. Among
these properties is the resistivity that decreases as the temperature of the wire increases and
hence its phase transforms to austenite. In the proposed control scheme (see Figure 6.1), SMA
electrical resistance (Rsma) is the only property measured, and the inner morphing-controller
is, in effect, servo’ing electrical resistance to follow a commanded profile. One might expect
that resistance change is related to motion change, however, there is no direct measurement
of motion to evaluate the conditions under which this Resistance-Motion (RM) relationship
is valid. This fact suggests to analyze how the RM is affected when both ambient (To) and
SMA temperatures (T ) change. Even small variations in temperature modify the RM function’s
slope. The following section details on this issue.
RM sensitivity upon temperature change
At ambient temperature (To = 22.7oC) the Resistance-Motion function satisfies the linear
equation:
q3 = 10(8.5−Rsma) (6.5)
The function in Eq. (6.5) is determined by measuring the changes in SMA electrical re-
sistance during the heating phase of the wire. By also measuring the elbow joint rotation is
possible to determine a relationship between resistance and angular motion. This resistance-
angle relationship is linear because the martensite fraction is kinematically coupled to the
rotation, and the martensite fraction is what drives the resistance changes. Figure 6.7 shown
the experimental results. While the applied voltage and current to the SMA both change with
the wire strain in a hysteretic fashion, the resistance of the SMA wire (Rsma) changes almost
linearly with the angular motion q3. Several experiments at ambient temperature confirm the
reliability of the linear function.
However, the airflow generated by the flapping motion of the wings and also the airspeed
during flight might cause the SMA temperature to decrease, thus changing the RM function.
In Figure 6.7 it has been observed that variations in temperature lead to changes in resistance
decrease rate. Interestingly, changes were only produced in the slope of the curve, remained
the function linear. This phenomenon enables to state the RM relationship satisfy the model:
131
6.5 Morphing-wing control (inner loop)
q3 =M−1(Rsma − b), being M the slope of the curve. Using this model is possible to estimate
the angular motion q3 by measuring Rsma and calculating the parameters M and b.
6.5 Morphing-wing control (inner loop)
Figure 6.8 expands the inner control loop previously shown in Figure 6.1. It details how the
heating power signal uheating is generated such as mentioned in step 5 of Algorithm 3, cf. Eq.
(6.6). Two mechanisms: anti-slack and anti-overload complement the morphing-wing controller.
Two control strategies have been developed and implemented as morphing controllers: i) sliding-
mode control with estimated force reference, and ii) PID position control. In the following, both
controllers will be derived and compared.
AntagonisticSMA mechanism of the wings
Morphing controller-+ [ref]
[Uheating]SMA actuation
++ Ƭ3
Anti-slack mechanism
Anti-overload mechanism
++
[feedback]
u_high
u_low
u3
Figure 6.8: General scheme for the inner morphing-wing control. Source: the author.
uheating = u3 + ulow + uhigh,∈ �2, (6.6)
In Eq. (6.6) u3 refers to the morphing-wing control signal and ulow, uhigh are the lower and
upper values of input power that are regulated by the anti-slack and anti-overload mechanisms
respectively. Both mechanisms are used for both sliding and PID controllers.
6.5.1 Sliding-mode control
Figure 6.9 expands the inner loop shown in Figure 6.8. It completely details the components
required for the sliding-mode control method. The reference represents a torque profile to drive
elbow motion (τ3,ref ). Mapping from torque-to-motion requires the inertial model (FD). To
derive the control law equation u3 using the sliding-mode foundation, the following procedure
is applied:
132
6.5 Morphing-wing control (inner loop)
-+
Fsma_1
Fsma_2-1
SMA actuation
Resistancemeasurement
-+
Sliding mode controller
MIN
-+
0.03W
0.95
anti-slack mechanism
MAX
-+
3W
1.25
anti-overload mechanism
++
++
u_high
u_low
0.016/(0.35s+1)
0.016/(0.35s+1)
AntagonisticSMA mechanism of the
wings
u3
FD [q3,ref]
Inerial Model
q3 = M −1 Rsma − b( )RM function
q3
uheating τ 3τ difτ 3,ref HT( )−1 I3 q3,ref + ′Ksmce+ HT I3−1ξ3 +αsat s( )⎡⎣ ⎤⎦
Rsma =VsmaIsma−1
Psma = Isma2 Rsmainput power
Figure 6.9: Detailed inner loop of sliding-mode morphing-wing control with torque reference.
Source: the author.
1. Define the position error: e = q3,ref − q3 and its dynamics e.
2. Define the sliding surface S = e+K ′smce. S defines the dynamics that governs the system
behavior while sliding. K ′smc > 0 corresponds to the control parameters vector.
3. The sliding control signal u3 is designed by ensuring the references slide along the S
surface. A Lyapunov function is defined as: V = 0.5ST , S > 0.
4. The sliding control is chosen such as V = STS < 0, or STS ≤ −α |S| = −αST sgn(S),being sgn() the sign function, and α a positive scalar.
5. The sliding condition is S = −αsgn(s).
By differentiating S = e+K ′smce with respect to time:
S = q3,ref − q3 +K ′smce (6.7)
In Eq. (6.7) the term q3 governs the angular acceleration dynamics of the elbow joint. By
using the inertial model from Algorithm 3, this term corresponds to:
q3 = HT[I−13 (F3 − ξ3)
], (6.8)
where the 6x1 vector H allows for the projection of the spatial force of the elbow F3 onto
the axis of motion, ξ3 contains Coriolis and gyroscopic forces, and I3 is the six-dimensional
inertia matrix. By substituting Eq. (6.8) into Eq. (6.7):
S = q3,ref −HT[I−13 (F3 − ξ3)
]+K ′smce (6.9)
133
6.5 Morphing-wing control (inner loop)
By equaling Eq. (6.9) with the sliding condition S = −αsgn(s), and then isolating the force
term F3:
−αsgn(s) = q3,ref −HT[I−13 (F3 − ξ3)
]+K ′smce
F3 = (HT )−1I3[q3,ref +K ′smce+HT I−13 ξ3 + αsgn(s)]
(6.10)
By renaming u3 = F3, the sliding control law is derived from Eq. (6.10), as:
u3 = (HT )−1I3[q3,ref +K ′smce+HT I−13 ξ3 + αsgn(s)] (6.11)
Finally, to avoid chattering phenomenon, i.e., oscillations when the system approaches the
sliding region, the sign function is replaced by the saturation function sat(S), as:
sat(S) =
{sgn(S), S ≥ 0S/φ, |S| ≤ 0
}(6.12)
By adding Eq. (6.12) into (6.11), the sliding mode control law is:
u3 = (HT )−1I3[q3,ref +K ′smce+HT I−13 ξ3 + αsat(s)] (6.13)
The term ξ3 is defined in Eq. (4.10). The control gains K ′smc and α should be positive
scalars. Section 6.7 details the parameters values used for simulation and experimental testing
of the morphing-wing control using sliding-mode technique.
6.5.2 PID control
Figure 6.10 expands the inner loop shown in Figure 6.8 based on PID position control scheme.
The reference is the angular position profile of the elbow joint (q3,ref ). Using the linear model
in Eq. (6.4), the PID controller has been tuned using the Ziegler-Nichols methodology. The
PID transfer function is given by Eq. (6.14).
u3 = Kp
(1 +Kis
−1 +Kds), (6.14)
Where PID constants are: Kp = 35, Ki = 0.006, and Kd = 0.08. PID control is very simple
and most importantly, easy to program using the onboard limited processing capabilities of
BaTboT. It does not require complex computation and calculation. However, what really
makes the PID strategy the best choice for morphing-wing control driven by Shape Memory
Alloys recalls on mechanisms that enhance the PID control law. These mechanisms are defined:
i) anti-slack, and ii) anti-overload.
134
6.5 Morphing-wing control (inner loop)
-+
Fsma_1
Fsma_2-1
Resistancemeasurement
-+
PID
MIN
-+
0.03W
0.95
anti-slack mechanism
MAX
-+
3W
1.25
anti-overload mechanism
++
++
u_high
u_low
0.016/(0.35s+1)
0.016/(0.35s+1)
AntagonisticSMA mechanism of the
wings
35(1+0.006/s+0.08s)u3
[q3,ref]
q3
uheating τ 3τ dif
Rsma =VsmaIsma−1
Psma = Isma2 Rsmainput power
q3 = M −1 Rsma − b( )RM function
SMA actuation
Figure 6.10: Detailed inner loop of PID morphing-wing control with joint position reference.
Source: the author.
6.5.3 Mechanisms
Both sliding-mode and PID based inner control loops make use of two mechanisms that com-
plement the control law: anti-slack and anti-overload, cf. Figures 6.9 and 6.10 respectively.
These mechanisms allow:
• Anti-slack mechanism: it defines a minimum threshold of input heating power Pmin, that
ensures the inactive SMA wire does not cool completely. This lower value of input heating
power is ulow.
• Anti-overload mechanism: It is in charge of ensuring that the maximum input power does
not increase above an upper limit Pmax. This upper value of input heating power is uhigh.
The anti-slack mechanism
As explained in (67), the purpose of the anti-slack mechanism is to deal with the two-way
shape memory effect, which is mainly produced when the SMA wires extend upon cooling. The
passive SMA wire can develop a few millimeters of slack as it cools, which consequently affects
the accuracy of the control. The two-way shape memory effect becomes even more problematic
in the antagonistic arrangement of SMA actuators, and may lead to slower response and even
wire entanglement.
To avoid slack issues, the anti-slack mechanism defines a minimum threshold of input heating
power Pmin, that ensures the inactive wire does not cool completely. The improvement in
135
6.5 Morphing-wing control (inner loop)
actuation speed is due to the fact that the already-warm SMA wire can begin to pull as soon as
the heating current is raised, whereas a cold wire would first need to be raised to its operating
temperature. The mechanism compares the minimum value of the input power Psma of each
SMA with Pmin, ensuring that this applied power does not drop below the lower limit. In
the current system Pmin = 0.03W . After several simulations and experiments carried out
in section 6.7, it was empirically determined that by keeping 10% of the maximum applied
electrical current on the inactive SMA actuator, the mechanism works as expected. Thereby,
ulow = KsP , where Ks = 0.95 of the gain of the mechanism.
The anti-overload mechanism
The anti-overload mechanism is in charge of ensuring that the maximum input power does not
increase above an upper limit, defined as Pmax =∼ 2.57− 3W (calculated from the maximum
allowed input electrical current I =∼ 550 − 600mA, which was previously set in Figure 4.22,
Section 4.6.4). This approach avoids overheating the SMAs in case the controller delivers a
large amount of power. This input power saturation is due to uhigh = KoP , where Ko = 1.25
of the gain of the mechanism. It is important to highlight that the gains of each mechanism
(Ks, Ko) have been experimentally obtained to allow the elbow joint to rotate at a maximum
speed of ∼ 0.5deg/ms (cf. Figure 6.5).
6.5.4 Morphing-wing control algorithm
Algorithm 3 details how the inner-loop that drives the morphing-wing motion of the wings
operate. It runs typically at 30Hz, generating 12 samples per wing-stroke (12f), being f =
2.5Hz. In the first place, closing the inner loop requires the feedback of elbow motion q3.
Thanks to the RM function is possible to estimate q3 during the wingbeat cycle (cf. Figure
6.7). Having q3 estimated, the morphing-wing controller can operate (by using the sliding-
mode or PID strategies). Algorithm 3 works as follows: The outer -while- loop handles the
wingbeat cycle of the wings: downstroke and upstroke. During the downstroke, the SMA2
actuator is active whereas SMA1 is passive, allowing the wings to extend i.e., rotation of
elbow’s joint from ∼ 60o to 5o, (cf. Figure 6.11). Contrary during the upstroke, SMA1
turns active which allows the wings to contract. In step 1, an initial measurement of SMA
electrical resistance is measured and buffered in Rsma,i. This step is repeated at the beginning
of each wing stroke. The inner -while- loop handles the activation of each SMA actuator. This
loop ends when the -contracted - pin of the SMA actuator turns active, indicating it is fully
136
6.6 Attitude control (outer loop)
——————————————————-
ALGORITHM 3: morphing-wing control
——————————————————-
Initialize k ← 1, i ← 1, Mi ← 0.1, bi ← 8.5
while i < #wingbeat cycles do
1. Measure and buffer Rsma,i
2. Rsma,k ← Rsma,i
while CONTRACTED! = 1 do
3. Compute and feed-back q3,k ←M−1
i (Rsma,k − bi)
4. Calculate position error q3,ref − q3,k5. Run morphing control as uheating,k ← u3,k+
ulow,k + uhigh,k
6. k ← k + 1
7. Measure Rsma,k
end while
8. Buffer Rsma,f ← Rsma,k
9. Calculate new slope Mi+1 ← Rsma,f−Rsma,i
60o
10. Calculate new bi+1 ← Rsma,f −Mi60o
11. i ← i+ 1, k ← 1
end while
downstroke (SMA_2 active) upstroke (SMA_1 active)i=1, k=1
i++, k=1
k++
Measure
CONTRACTED =1
Measure
SMA_1
SMA_2
q 3,refdeg
[]
t s[ ]
Rsma,iRsma, f
Figure 6.11: Typical elbow joint reference
profile q3,ref during a wingbeat cycle (f =
1.25Hz). It details how Algorithm 3 works.
contracted. Inside this loop, at k = 1 (step 3), ebow joint is estimated by the RM model:
q3,k ← M−1i (Rsma,k − bi), where initial values are set to M = 0.1 and b = 8.5. Estimated
values of q3,k are feed-back to the control module aimed at calculating the position error of the
elbow joint: q3,ref−q3,k (step 4). The control signal uheating is generated by the morphing-wing
controller in step 5. uheating is a contribution of: morphing-wing control signal (u3), anti-slack
mechanism (ulow) and anti-overload mechanism (uhigh). Steps from 3 to 7 are repeated until
the end of each wingstroke, where new values of slopeM and term b are calculated based on the
final measurement of electrical resistance Rsma,f (steps 8 to 11). Simulations and experimental
results of the morphing-wing control can be found in section 6.7.
6.6 Attitude control (outer loop)
In bats, there is biological evidence that the inertial forces produced by the wings have a
significant contribution into the attitude movements of the animal, even more significant than
aerodynamic forces (24). In fact, bats perform complex aerial rotations by modulating solely
wing inertia (26). This means bats are able to change the orientation of the body during flight
without relying on aerodynamic forces and instead by changing the mass distribution of its
body and wings. Inertial forces are likely to be significant in bats because the mass of the wings
comprises a significant portion of total body mass, ranging from 11% to 33% and because wings
137
6.6 Attitude control (outer loop)
undergo large accelerations (49).
1
2
3
4
1
2
3
4
(a) (b)
3D joint points
Figure 6.12: a) In-vivo recordings of a bat land-
ing on the ceiling, cf. (26), b) closed-loop control
simulation of attitude maneuvering using the back-
steping+DAF strategy. Source: the author.
Thus, taking into account the effects
of wing inertia within the control law is
a key factor for the design of an attitude
controller of the proposed robot. The ap-
proach followed is aimed at defining proper
references that drive the modulation of the
wings’ shape in such a way to increment
inertial forces that generate propulsion. To
achieve this, an attitude control strategy
based on backstepping methodology plus
a function (DAF) to produce desired roll
and pitch angular accelerations are used.
The DAF function contains wing inertia
information that is provided by the iner-
tial model computation of the Equations of
Motion (EoM) of the robot. This enhanced
controller is called backstepping+DAF.
The novelty of the controller is: It pro-
poses a nonlinear control approach,
called backstepping+DAF, aimed at
improving the attitude response of the
MAV. Such enhancement is based on
the assumption (motivated by the cited biological studies in Chapter 3) that bats
efficiently generate forward thrust by means of inertia wing modulation, taking ad-
vantage of relevant wing-to-body mass ratio.. Figure 6.12 compares aggressive attitude
maneuvers executed by real bats (plot-a) and those obtained by the backstepping+DAF con-
troller from the simulator (plot-b). Joint trajectories of reference in plot-a has been extracted
from the specimen using high speed cameras (cf. (26)), and those references have been used by
the SimMechanics closed-loop simulator, cf. Annex 10.6, Figure 10.2. Annex 10.6 details the
SimMechanics closed-loop environment which is based on the open-loop environment previously
introduced in Figure 4.12. It complements the open-loop environment by adding both inner-
loop morphing-wing control and outer-loop attitude control. Because morphing-wing control
138
6.6 Attitude control (outer loop)
BacksteppingMotor
outer loop: attitude control R/C (flapping)
IMU
BaTboT model
+-
R/C (attitude)forward/turning
DAF
inner loop: morphing control
[q1…q6]R,L
wingtrajectories
[roll, pitch]
q1,refφ d ,θ d⎡⎣ ⎤⎦φ,θ[ ]
uφ ,uθ⎡⎣ ⎤⎦
wristmapping
φ,φ,θ ,θ⎡⎣ ⎤⎦
uθ = q2,ref
uφ = q3,ref
-+
Fsma_1
Fsma_2-1
Resistancemeasurement
-+
PID
MIN
-+
0.03W
0.95
anti-slack mechanism
MAX
-+
3W
1.25
anti-overload mechanism
++
++
u_high
u_low
0.016/(0.35s+1)
0.016/(0.35s+1)35(1+0.006/s+0.08s)
u3
q3
uheating τ dif
Rsma =VsmaIsma−1
Psma = Isma2 Rsmainput power
q3 = M −1 Rsma − b( )RM function
SMA actuation
τ1τ 2
τ 3τ 4τ 5τ 6
Figure 6.13: Detailed outer and inner loops: complete description of the Flight Control Archi-
tecture (FCA). Source: the author.
can be based on sliding-mode technique or PID technique, the former is fully described in Fig-
ure 10.2 whereas de latter in Figure 10.2. The SimMechanics closed-loop environment
is not only used for simulation purposes but also for experimental testing. It sends
and receives commands to/from the Arduino-nano board that is placed on the robot. Details
on software-to-hardware control interplay can be found in Figure 6.14. In addition, Figure
6.13 expands the outer-loop attitude control from Figure 6.1 and shows the Flight Control
Architecture (FCA) is more detail.
6.6.1 Backstepping+DAF
DAF: Desired Angular acceleration Function (φd, θd)
To formulate the DAF terms (φd, θd), the attitude data (θ, θ, φ, φ) that is feedback by the IMU
and the wing joint trajectories of reference (q) are required, as shown in Figure 6.13. Therefore
DAF terms can be written as a function of attitude and wing modulation data as:
φd = f(φ, φ, q, q)
θd = f(θ, θ, q, q)(6.15)
The desired roll and pitch angular accelerations (φ, θ) are components of the six-dimensional
body accelerations Vb that are produced by the inertial forces FT . The definition of Vb with
139
6.6 Attitude control (outer loop)
respect to the body frame {b} requires the computation of the inertial model from Algorithm
1, cf. step 5, and yields:
Vb =
(Ib +
[R0,b
0∑i=6
(Ri+1,iPTi,i+1Ii,R)+R0,b
0∑i=6
(Ri+1,iPTi,i+1Ii,L)
])−1
FT (6.16)
In Eq. (6.16), Ib is the spatial inertia of the robot’s body calculated with respect to the
body frame {b}, whereas the term0∑i=6
Ii expresses the propagation of wing inertias of both
wings onto the base frame {0}. Subscripts R and L refer to the right and left wing respectively.
The term R0,b applies a rotation to express wing inertias with respect to the body frame {b}.Finally, DAF terms are defined and expressed with respect to the body frame {b} as:
φd =[1 0 0 0 0 0
]Vb,
θd =[0 1 0 0 0 0
]Vb
(6.17)
Backstepping+DAF control
This section shows roll control derivation for uφ. Pitch control derivation (uθ) follows the same
procedure. The first step is to define the roll tracking error e1 and its dynamics (derivative
with respect to time):
e1 = φd − φ
e1 = φd − ωy(6.18)
The term φd corresponds to the desired roll trajectory profile, whereas φ is the roll angle
measured by the IMU sensor. A positive definite Lyapunov function (L) is used for stabilizing
the tracking error e1, as:
L (e1) =e212 , (6.19)
To regulate the behavior of the angular velocity ωy from e1 in Eq. (6.18), a second tracking
error e2 is defined, as:
e2 = ωdy − ωy (6.20)
The desired behavior for e2 is defined as:
ωdy = c1e1 + φd + λ1∫e1 , (6.21)
140
6.6 Attitude control (outer loop)
where c1, λ1 are positive constants and∫e1 is the integral of the roll tracking error. In
other words, ωdy is considered as a virtual control law that governs the behavior of e2. Now,
substituting ωdy into Eq. (6.20) and differentiating e2 with respect to time (note that ωy = φ):
e2 = c1e1 + φd + λ1e1 − φ (6.22)
From Eq. (6.20) ωy = ωdy − e2. Substituting ωy into e1 from Eq. (6.18) gives:
e1 = φd − ωdy + e2 (6.23)
Now, substituting ωdy from Eq. (6.21) into (6.23):
e1 = φd − (c1e1 + φd + λ1∫e1) + e2
e1 = e2 − c1e1 − λ1∫e1
(6.24)
Finally, substituting e1 from Eq. (6.24) into Eq. (6.22):
e2 = c1(e2 − c1e1 − λ1∫e1) + φd + λ1e1 − φ (6.25)
By expressing φ in terms of the pitching torque τφ and the moment of inertia of the robot’s
body Ixx about the x axis of the body frame {b} φ = τφ/Ixx and defining uφ = τφ, Eq. (6.25)
can be written as:
uφ = Ixx[c1(e2 − c1e1 − λ1∫e1) + φd + λ1e1 − e2] (6.26)
Replacing e2 = −e1 −λ2e2 into Eq. (6.26), the backstepping+DAF control law in charge of
roll regulation is:
uφ = Ixx[e1(λ1 − c21 + 1) + e2(c1 + λ2)− c1λ1∫e1 + φd] (6.27)
The parameters of the controller, λ1, c1, λ2 > 0 are defined in Table 6.2. By applying the
same procedure, the backstepping+DAF control law in charge of pitch regulation is given by:
uθ = Iyy[e3(λ3 − c22 + 1) + e4(c2 + λ4)− c2λ3∫e3 + θd] (6.28)
By substituting the DAF terms φd, θd into Eq. (6.27) and (6.28):
uφ = Ixx[e1(λ1 − c21 + 1) + e2(c1 + λ2)− c1λ1∫e1 +
[1 0 0 0 0 0
]Vb
uθ = Iyy[e3(λ3 − c22 + 1) + e4(c2 + λ4)− c2λ3∫e3 +
[0 1 0 0 0 0
]Vb
(6.29)
141
6.6 Attitude control (outer loop)
Stability Analysis
The following candidate Lyapunov function has been chosen:
L = 0.5(e21 + e22 +∫e1
2) (6.30)
Differentiating Eq. (6.30) with respect to time, having e1 = e2 − c1e1 − λ1∫e1 and e2 =
−e1 − λ2e2 gives:
L = e1e1 + e2e2 + e21 ≤ 0= e1(e2 − c1e1 − λ1
∫e1) + e2(−e1 − λ2e2) + e21
= −c1e21 − λ2e22 ≤ 0
(6.31)
The fact that Eq. (6.31) fulfils L ≤ 0, ∀(e1, e2) ensures the boundedness of e1, its integral∫e1, and e2. Hence, the reference angular value φd and the roll angle φ are also bounded
due to e1 = φd − φ. The boundedness of e1 implies that the virtual law ωdy is bounded as
well, which consequently makes the error dynamics e2 and the DAF term φd also bounded.
Furthermore, global asymptotic stability is also ensured due to the positive definition of L, in
which L(e1, e2) < 0, ∀(e1, e2) = 0, and L(0) = 0 (by applying LaSalle’s theorem).
6.6.2 Attitude control algorithm
The attitude control procedure based on the proposed backstepping+DAF methodology is
detailed in Algorithm 4. The algorithm is based on the Flight Control Architecture (FCA)
detailed in Figure 6.13.
ALGORITHM 4: Backstepping+DAF attitude control
——————————————————-
1. Read filtered data from IMU: φ, φ, θ, θ
2. Compute Vb from Algorithm 1 using IMU readings.
3. Calculate DAF terms: φd, θd
4. Run backstepping+DAF control: uφ, uθ
5. Run inner-loop morphing-wing control Algorithm 3.
The computation of the inertial model from Algorithm 1 is required for calculating the DAF
terms that compose the attitude controller. The inputs to the inertial model are the wing joint
trajectories of reference (q, q, q) and the filtered attitude IMU measurements (φ, φ, θ, θ). The
output of the inertial model is the spatial body accelerations Vb that are produced by wing
forces that act on the center of mass of the robot. DAF terms are then extracted from Vb in the
form of desired angular acceleration commands which drive roll and pitch behaviour (φd, θd).
142
6.7 Simulation and experimental results
Both DAF terms are added into the backstepping control law aimed at improving roll and pitch
regulation (uφ, uθ). Improvements are expected in terms of increments of inertial thrust due
to more efficient modulation of the wings that are driven by the backstepping+DAF control
outputs. The following section presents preliminary results of the Flight Control Architecture
(FCA) on BaTboT.
6.7 Simulation and experimental results
Simulations are carried out using the closed-loop Simmechanics Matlab environment shown
in Annex 10.6, whereas experiments are conducted by using the closed-loop setup described
in Figure 6.14. The setup shows the software and hardware interplay of the Flight Control
Architecture (FCA) modules from Figure 6.13.
16-bit ADC + anti-aliasing filter9620-05-ATI
wind-tunnel testbed withforce sensor load
SMA driver
arduino-nano SMA resistancemeasurement
FT
user
Matlabenvironment
[u1]
[u3,R u3,L]
(morphing)
[uheating,L,R]
[u1]
(flapping)
[Rsma,L,R]
[roll pitch ]φ,φ,φ,θ ,θ ,θ
[R_feedback]
[Isma]
IMUφ θ
(force)
Figure 6.14: Experimental setup using the wind-tunnel of Brown University: Flight Control
Architecture (FCA). Source: the author.
BaTboT is equipped with onboard processor based on Arduino technology which uses a
PIC18F2680 that mainly receives the commands from the external PC (Matlab environment)
via serial connection. These commands are the control outputs for: flapping u1 and morphing-
wing modulation u3,R,L (subscripts R and L refer to the right and left wing respectively).
The SMA driver is based on a MOSFET transistor that receives the control command u3
and generates the driving electrical current signal to operate each SMA actuator of the wings.
143
6.7 Simulation and experimental results
BaTboT is also equipped with onboard IMU that feedbacks attitude measurements: φ, φ, θ, θ.
These signals are filtered inside the Arduino processor using a Kalman filter before feedback to
the PC. Also, SMA electrical resistance changes are also measured (Rsma) and feedback to the
PC. Finally, a cell load within the wind-tunnel equipped with a force sensor allows for getting
measurements of 6D forces produced at the center of mass of the robot. A 16-bit DAC with
embedded 1st order anti-aliasing filter is used for that purpose.
Attitude outer loop runs at 20Hz based on IMU readings, whereas morphing-wing inner loop
runs at 30Hz based on SMA resistance change readings. Details on these loops are depicted in
Figure 6.13.
6.7.1 Variables and parameters
The list of parameters used for both simulation and experimental testing are consigned in Table
6.2. It details morphological, modeling and control parameters.
Table 6.2: List of robot’s parameters: morphological, modeling, control.
Parameter (unit) Value
Body mass Mb (g) 125
Extended wing length B (m) 0.245
Body width lm (m) 0.04
Body inertia tensor diagonal [Ixx, Iyy , Izz ] (gcm2) [1, 0.07, 0]
Extended wing spanS = lm + 2B (m) 0.53
Extended wing area Ab (m2) 0.05
Humerus lengthlh (m) 0.055
Humerus inertia tensor diagonal [Ixx, Iyy , Izz ] [gcm2] [0.03, 0.37, 0.93]
Humerus position vector to CM s2,cm (m) [0.0275, 0, 0]
Radius lengthlr (m) 0.070
Radius inertia tensor diagonal [Ixx, Iyy , Izz ] (gcm2) [0.07, 0.92, 0.37]
Radius position vector to CMs3,cm (m) [0.035, 0, 0]
Air density ρair (Kg/m3) 1.2
Lift coefficient CL at angle of attack AoA = 9o 1.5
drag coefficientCDat angle of attack AoA = 9o 0.152
Average lift force FL at angle of attack AoA = 9o (N) 0.97
Average drag force FD at angle of attack AoA = 9o (N) 0.099
Sliding mode morphing-wing controller parameters K′smc, α 2.45, 1.1
PID morphing-wing controller parameters Kp, Ki, Kd 35, 0.006, 0.08
Anti-slack mechanism gain Ks 0.95
Anti-overload mechanism gain Ko 1.25
Backstepping+DAF attitude control (roll) [λ1, c1, λ2] [1.87, 2.1, 0.02]
Backstepping+DAF attitude control (pitch)[λ3, c2, λ4] [4.5, 2.5, 0.02]
144
6.7 Simulation and experimental results
6.7.2 Morphing-wing control response
In this section both derived sliding-mode and PID methods are compared aimed at determining
the best performance for driving SMA actuators. As introduced at the beginning of this chapter
the control goal relies on achieving the fastest wing modulation via SMA actuation
without compromising power consumption and yet achieving a feasible SMA output
torque. Both control strategies are evaluated in that regard.
Sliding-mode response
The control law derived in Eq. (6.13) whose diagram block is detailed in Figure 6.9 is first tested
on simulation. Figure 6.15 shows the results. Sliding-mode control was initially proposed
under the thought of being capable of driving faster SMA contraction thanks to the energy
management ensured by the control law. With that in mind, a sinusoidal joint position reference
for elbow rotation (q3,ref ) was set to a frequency of 4Hz. Plot-a in Figure 6.9 shows the tracking
results. A first problem is observed in terms of error tracking which causes a delay in tracking
the reference. Several simulation results have shown similar response. The problem is caused
by the saturation function of the control law (cf. Eq. 6.12). This function avoids chattering
phenomenon, which means oscillations when the system approaches the sliding region of the
surface S, and shown in plot-b. This technique does not ensure the asymptotic stability of the
system and compromises an accurate response by smoothness. As a consequence plot-a shows
a smooth tracking but unfortunately a large tracking error.
In terms of power consumption the sliding-mode does well in simulation. Plot-c in Figure
6.15 details the driven current consumed by each SMA actuator. Note how the anti-slack
mechanism ensures the input heating power does not drop below the lower threshold of 0.03W ,
which in terms of electrical current corresponds to 0.06A. This approach allows to keep the
SMA wires away from completely cooling which in turns causes that the wires can respond
faster under activation/contraction. Using the sliding-mode technique a maximum current of
0.2A is required to actuate the SMA at the driven frequency. This is a good result taking
into account each SMA contracts in 100ms. In Figure 6.16 experiments are conduct aimed at
verifying the simulation results.
As mentioned in Section 6.1, one of the control goals is to evaluate efficiency as a tradeoff
between SMA output torque (τ3) and input power (Psma = I2smaRsma). By using the inertial
model to map torque references (τ3,ref ) to joint position references (q3,ref ) and by measuring
145
6.7 Simulation and experimental results
(a)
q3 [rad] -1 -0.5 0 0.5 1 1.5
-2
q3 [r
ad/s
]
.
-1
0
123
4
5
6
S=0
t [s]
S
e
e.
sliding surfacesat(S) function
0t [s]
0.2 0.4 0.6 0.8 1
Isma [
A]
0
0.05
0.1
0
0.15
0.2
0
Isma_1Isma_2
(b)
q3
q3,ref
0 0.1 0.2 0.3 0.4 0.5 0.6t [s]
-1
-0.5
0
0.5
1
(c)
q3 [r
ad]
Anti-slack mechanism(Imin=0.06A)
Figure 6.15: (Simulation) morphing-wing based on sliding-mode response: a) control tracking
given a sinusoidal joint trajectory (q3,ref ) at wingbeat frequency of f = 4Hz, b) phase plane of
the sliding surface upon sinusoidal input from plot (a), c) electrical current (Isma) to drive each
SMA actuator in the antagonistic configuration. Source: the author.
the output torque (τ3) using the setup shown in Figure 6.2, experimental results regarding
torque tracking are depicted in Figure 6.16a. It turns out the sliding-mode technique causes
fatigue issues on the SMA wires due to large peaks of current delivered. This can be clearly
observed in plot-a. Note how during the second wingbeat (t > 0.6s) the output torque τ3
saturates even under the presence of electrical heating. This causes large errors in tracking that
can be appreciated in Figure 6.16b. The saturation of output torque occurs due to a physical
decrease of performance of the SMAs which in turns limits the angle range of motion of the
elbow joint q3. At the beginning of the experiment it was thought that these large errors in
joint tracking were caused by the saturation function of the sliding-mode control, similarly with
those observed from the simulations, cf. Figure 6.15a. Nonetheless, after removing the sat(S)
function from the control law, the tracking error did not improved. Figure 6.16b compares the
joint position tracking with and without the saturation function. Therefore, it was discovered
that fatigue issues on the SMAs caused by the controller was the cause of these errors. Plot-b
contains the joint tracking curve of a wingstroke corresponding to the interval between 0.6s and
1s, which fatigue issues are clearly observed from the plot-a. As mentioned fatigue is caused by
146
6.7 Simulation and experimental results
Isma_1Isma_2
Isma [
A]
t [s]
τ3 [N
m]
τ3,ref
τ3
(a)
q3 [d
eg]
0 0.2 0.4 0.6 0.8 1 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 00.75 0.8t [s]
(b)
0
0.02
-0.02
0.01
-0.01
t [s]
0
12
24
36
48
60q3 with sat(S)q3 without sat(S)
q3,ref
(c)
Fatigue
0
0.25
0.5
0.75
1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
τ3,ref
τ3
Figure 6.16: (Experimental) morphing-wing based on sliding-mode response: a) SMA output
torque τ3 given a force reference τ3,ref , b) control tracking given a bio-inspired joint trajectory
q3,ref at wingbeat frequency of f = 2.5Hz; it shows comparison of sliding-mode response with
and without the saturation function sat(S) within the control law, c) electrical current (Isma) to
drive each SMA actuator in the antagonistic configuration Source: the author.
large applied electrical currents delivered by the controller. Plot-c describes the current profile
applied to each SMA actuator of the antagonistic arrangement. It corresponds to the first
wingstroke from 0s to 0.4s. Peaks up to 1A causes the SMA to fatigue. This causes an instant
decrease of output torque (cf. plot-a) and therefore a decrease in motion range. The decrease in
motion range is clearly observed in plot-b, causing the tracking error mentioned before. Fatigue
might cause a fatal damage of the shape memory effect. Further simulations and experiments
regarding sliding-mode control can be found in (97), whereas PID control response is presented
in the following.
147
6.7 Simulation and experimental results
0 0.5 1 1.5 20
20
40
60
t [s]
q 3[deg]
0 0.5 1 1.5 20
0.2
0.4
0.6
t [s]
I sma[A
]
0 0.5 1 1.5 20
20
40
60
t [s]
q 3[deg]
0 0.5 1 1.5 20
0.2
0.4
0.6
t [s]I sm
a[A]
0 0.5 1 1.5 20
20
40
60
t [s]
q 3[deg]
0 0.5 1 1.5 20
0.2
0.4
0.6
t [s]
I sma[A
]
q3,refq3
q3,refq3
q3,refq3
Isma,1Isma,2
Isma,1Isma,2
Isma,1Isma,2
Anti-slack mechanism(Imin=10%*Imax)
Anti-overload mechanism
(a) (b)
Figure 6.17: (Simulation) morphing-wing based on PID response: a) PID control tracking given
a sinusoidal profile q3,ref (above), square profile (medium), sawtooth profile (below), at wingbeat
frequency of f = 2.5Hz, b) electrical current (Isma) to drive each SMA actuator in the antagonistic
configuration. Source: the author.
PID response
PID control architecture was previously shown in Figure 6.10. The control law u3 in Eq. (6.14)
was tuned based on experimental identification of the plant, which in this case corresponds to
the SMA actuators. The identified plant were represented by a LaPlace first order function
shown inn Eq. (6.4).
The main advantage of the PID control relies on the less amount of power that delivers
to the SMAs compared to the sliding-mode. Firstly simulations are carried out to analyse
the tradeoff between joint tracking accuracy and delivered power. Figure 6.17a details the
simulation results taking into account three different profiles for describing the joint reference
q3,ref : sinusoidal, square and sawtooth. Figure 6.17b shows the corresponding input heating
current delivered by the PID controller. For instance it can be observed that the maximum
required current to drive both SMA actuators is always below 550mA, about the half than the
experiments carried out with the sliding-mode technique (cf. Figure 6.16c). In terms of tracking
the PID does well, specially for sinusoidal input references (cf. Figure 6.17a, above). Comparing
against the sinusoidal tracking achieved with the sliding-mode in Figure 6.15a, which phasing
148
6.7 Simulation and experimental results
0 2 4 6 8
0.005
0.01
0.015
0.02
0.025
t [s]
3 [Nm]
0 2 4 6 80
0.15
0.3
0.45
0.6
0.75
t [s]
I sma [A
]
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
12
24
36
48
60
72
t [s]q 3 [d
eg]
q3,refq3
Anti-slack mechanism(Imin=10%*Imax) 0
Anti-overload mechanism(Imax=0.6A)
downstroke upstroke
0
(a)
(c)
(b)
f=2.5Hz
Figure 6.18: (Experimental) morphing-wing based on PID response: a) measured SMA output
torque τ3, b) PID control tracking given a bio-inspired joint trajectory q3,ref , c) electrical current
(Isma) to drive each SMA actuator in the antagonistic configuration. Source: the author.
problems are presented, the PID is able to accurate track the reference with a position error of
1.5%. In addition, there has not been observed any issues related to fatigue problems. However,
further experiments carried out in Chapter 7 section 7.2.1.2 will assess and discuss this issue in
more detail.
Experimental response of PID morphing-wing control is detailed in Figure 6.18. Note in
plot-a the measured output torque is maintained thanks to the proper regulation of the driven
current. Plot-c depicts the current profile in where is clearly appreciated the effects of both
anti-slack and anti-overload mechanisms. The anti-slack monitors the driven current does not
drop below 10% the maximum threshold aimed at speeding up the actuation frequency. In
this experiment the frequency is about 2.5Hz. On the other hand the anti-overload mechanism
monitors the driven current does not increase above the maximum threshold aimed at avoiding
fatigue issues. This is another advantage of PID control. Note how simulations reported
in Figure 6.17b (above) and experiments in Figure 6.18c show similar applied current limits
(below 600mA) that enable the proper wing modulation of joint q3. Contrary to the PID
control, the sliding-mode cannot take advantage of the anti-overload mechanism to ensure the
current limitation because it saturates the control output avoiding proper control tracking.
149
6.7 Simulation and experimental results
Without the anti-overload mechanism fatigue issues may occur, as shown in cf. Figure 6.16a.
Finally Figure 6.18b details the PID control tracking of the elbow joint reference q3. PID is able
to track the reference with neglectable error. Tracking errors are presented because the motion
of the elbow that is feedback to the controller has not been directly measured, instead and
estimation of the motion based on the change of SMA electrical resistance is the only variable
measured (cf. Algorithm 3: RM function). This estimation contains accumulative errors that
are presented during the elbow tracking, nonetheless these errors are not significant to decrease
the performance of the PID controller.
6.7.3 Attitude control response
This section only presents preliminary results of backstepping+DAF control in terms attitude
tracking. Attitude control results regarding the goal of incrementing net body forces thanks to
the wing modulation driven by the backstepping+DAF control law will be covered and discussed
in the main experimental results Chapter 7 section 7.4.
Backstepping has been widely applied to robust flight control problems (98). The key idea
of the backstepping design is to benefit from the desired dynamic state feedback that composes
the control law u = f(z, zd) (99). In this case, z corresponds to (φ) or (θ) respectively. The
backstepping does not only ensure global asymptotically stabilisation of φ and θ, it also allows
for the definition of a Desired angular Acceleration Function (DAF) to define the terms (φd)
and (θd). Basically two terms within the backstepping+DAF are key for attitude control: i)
integral action, and ii) DAF terms. Both can be observed from Eq. (6.29), where integral
action is represented by the term c1λ1∫e1 and DAF terms are φd, θd. The attitude error e1 is
expressed by e1 = φref − φ in the case of roll regulation (similarly with pitch). Integral action
is key for stabilisation purposes, whereas DAF terms are key for tracking purposes. Both are
combined into a single control law that enables BaTboT with accurate attitude performance.
Figures 6.22 and 6.20 show simulation and experimental respond of backstepping+DAF
control for stabilisation purposes. In Figure 6.22, both roll and pitch angles must be stabilise
to zero. Note roll angle (φ) has been set to 20o whereas pitch angle (θ) to −20o. The controller
is able to regulate the roll angle with a maximum control effort about 0.03Nm and the pitch
angle with a control effort about 0.15Nm. Comparing both values against the experimental
measurements carried out for the quantification of roll and pitch maneuvering from Figure 4.19,
one can note the similarities between simulation and experimental data. Refer to Table 4.7 for
numerical values at f = 1.3Hz. Experimental values indicate 0.04Nm for roll and 0.11Nm for
150
6.7 Simulation and experimental results
0 2 4 6 8 10−20
−15
−10
−5
0
5
t [s]
θ[d
eg]
0 2 4 6 8 10−0.04
−0.03
−0.02
−0.01
0
0.01
t [s]
uφ [
Nm
]
0 2 4 6 8 10−0.05
0
0.05
0.1
0.15
0.2
t [s]
uθ [
Nm
]
0 2 4 6 8 10−5
0
5
10
15
20
t [s]
φ [
deg
]
uφ
uθ
φref
φ
θref
θ
Figure 6.19: (Simulation) backstepping+DAF attitude stabilization. Roll (φref ) and pitch (θref )
references are set to zero while initial attitude position is set to 20o and −20o for roll and pitch
respectively. Control efforts are also shown. Source: the author.
0 2 4 6 8−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
t [s]
φ [
deg
]
0 2 4 6 8−3
−2
−1
0
1
2
3
4
t [s]
θ [
deg
]
θref
θ
φref
φ
Figure 6.20: (Experimental) backstepping+DAF attitude stabilization. Roll (φref ) and pitch
(θref ) references are set to zero. The wind-tunnel airspeed has been set to 2ms−1 and the controller
must keep both angles close to zero. Source: the author.
pitch. In Figure 6.20, both roll and pitch angles must be stabilise to zero in spite of disturbances
produced by the wind-tunnel airspeed, which it has been set to 2ms−1. The oscillations around
the set points are caused by the natural wingbeat motion (f = 1.5Hz) and the aerodynamic
loads. Position errors less than 3o are observed from the experiment.
In terms of tracking, backstepping+DAF also does well. Figure 6.21 show simulation results
151
6.7 Simulation and experimental results
0 0.5 1 1.5 2 2.5 3−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
t [s]
φ [
deg
]
0 0.5 1 1.5 2 2.5 3−6
−4
−2
0
2
4
6
t [s]θ
[deg
]
φref
φ
θref
θ
Figure 6.21: (Simulation) backstepping+DAF attitude tracking. Roll (φref ) and pitch (θref )
sinusoidal references are tracked. Source: the author.
0 1 2 3−2
−1
0
1
2
φ [
deg
]
t[s]
0 1 2 3−10
−5
0
5
θ [
deg
]
t[s]
0 1 2 30
0.01
0.02
0.03
erro
r
t[s]
φref
at 0ms−1
φ at 0ms−1
φref
at 5ms−1
φ at 5ms−1
θref
at 0ms−1
θ at 0ms−1
θref
at 5ms−1
θ at 5ms−1
φ error at 5ms−1
φ error at 0ms−1
0 5 10 150
0.005
0.01
0.015
0.02
erro
r
t[s]
θ error at 5ms−1
θ error at 0ms−1
1 2 3
Figure 6.22: (Experimental) backstepping+DAF attitude tracking. Roll (φ) and pitch (θ) pro-
files are tracked for wind-tunnel airspeeds of 0 and 5ms−1. Tracking errors are measured. Source:
the author.
regarding roll (φref ) and pitch (θref ) sinusoidal references. Tracking errors are quantified during
the experiments shown in Figure 6.22, which also depicts attitude tracking for different wind-
tunnel airspeeds profiles; 0 and 5ms−1 respectively. Tracking errors less than 3% are observed.
152
6.8 Remarks
6.8 Remarks
Regarding morphing-wing control, PID has demonstrated to be sufficient for achieving accurate
SMA actuation. It enables the SMA actuators to actuate at a maximum frequency of 2.5Hz
while ensuring accurate tracking and proper power consumption. Similar conclusions can be
found in (9). In fact, before (9) and (67), PID controllers were considered slow and inefficient,
and investigations regarding PID with high gains were never carried out. In this thesis, we have
also demonstrated that the power-to-force AC response of a NiTi SMA wire can be accurate
modeled as a first order low-pass filter, which transfer function was experimentally identified
by carrying out a frequency response analysis over the range of interest. This model is shown
in Eq. (6.4). Because the AC response of the SMA muscles resemble a first order low-pass
filter, the use of a PID seems to be an adequate solution. It is easy to program it onboard the
robot’s microprocessor and it can also be easily tuned under both simulation and experiments.
Basically, the main advantage of the PID control relies on the less amount of power that delivers
to the SMAs thanks to the incorporation of both mechanisms that monitors upper and lower
thresholds of SMA input power.
Regarding attitude control, the proposed backstepping+DAF controller has demonstrated
to be key for achieving proper response in terms of attitude stabilisation and tracking. Integral
action cancel steady errors whereas DAF terms improve on tracking response. However, the
key role of the DAF terms is to include wing inertia information that enables proper wing mod-
ulation. This issue has not been evaluated in this section but it will be completely introduced
in the following Chapter 7.
153
7
General experimental results
The fastest maneuvers in flying animals could be reproduced
in man made flying vehicles.
7.1 Overview
This chapter presents further experiments aimed at discussing the methods (modelling and con-
trol) proposed in this thesis and assessing their potential for developing bio-inspired morphing-
wings bat-like aerial robots. Along the previous chapters, it is has been presented a detailed
workflow that describe all the processes involved in that goal, from the analysis of biological
data that inspire and define the design of BaTboT, to the kinematics, dynamics, aerodynamics,
actuation and control methods that enable BaTboT to behave like its biological counterpart.
7.1.1 Methods and goals
Experiments are categorised in three areas:
1. Control performance. It evaluates both morphing-wing and attitude control response.
The goal of the experiments is twofold: (i) to assess the performance of SMA actuators
in terms of accuracy, limitations and impact into the proper modulation of the morphing-
wings. (ii) to evaluate how the proposed attitude controller enables accurate forward and
turning flight.
2. Aerodynamics. The goal is to demonstrate the impact of both morphing-wing and attitude
controllers into the proper generation of aerodynamic forces. It also discusses how to
154
7.1 Overview
robotic-arm(angle of attack)
high-speed camera
6D force sensor
high-speed camera
wind-tunnel mount-platform
(a)
(b)
Wind-tunnel test section
Figure 7.1: a) setup for morphing-wing testing. b) wind-tunnel setup for dynamics, aerodynamics
and control testing. Source: the author.
induce accurate aerodynamic behaviour for similar robots, not necessarily bat-like, based
on kinematics, dynamics and control parameters.
3. Efficient Flight. Efficiency is evaluated in terms of net force production. The goal is to
quantify how net forces can be increased by modulating wing inertia in a proper way.
Here, the hypothesis introduced at the beginning of this thesis is demonstrated, which
states that by including wing inertia information into the attitude controller, efficient
wing modulation that induces proper attitude behaviour is achieved.
155
7.2 Control performance
7.1.2 The wind-tunnel setup
Quantification of dynamics and aerodynamics data requires a complex setup. For this, the
Brown University’s wind-tunnel facility is used. Along the chapters of this thesis, a brief
description of the wind-tunnel components have been presented, as shown in Figure 6.14, how-
ever, a detailed description of the entire setup has not been shown. To this purpose, Figure
7.1 describes the main experimental setups used in this thesis. Firstly, the design of a highly
articulated morphing-wing required the use of a setup specially conceived for assessing SMA ac-
tuation. This setup is depicted in Figure 7.1a and enables half-part of the robot to be mounted
in (right wing). Secondly, Figure 7.1b shows the wind-tunnel setup. It enables the entire robot
to me mounted on a 6D force sensor that measures dynamics and aerodynamic forces. Wind
tunnel experiments have been conducted at the Brown University Breuer’s lab facility, which
is a closed-loop circuit with a test section measuring 3.8m in length and a cross-section of 0.60
by 0.82m (height X width). The wind-tunnel maximum airspeed is 20ms−1 and the stream
turbulence level of the tunnel is quite low (0.29% at 2.81ms−1). High-resolution CMOS cam-
eras (Photron 1024 PCI, resolution 1024x1024 pixels, lens 85mm, f/1.4) allow to capture all
markers useful for the kinematics extraction.
7.2 Control performance
This section complements experimental tests carried out in Chapter 6. The list of parameters
used for the following experiments remains the same than those introduced in Table 6.2. Also,
PID morphing-wing inner control loop and backstepping+DAF attitude outer control loop
structures remain the same than those introduced in Figure 6.13.
7.2.1 Morphing-wing control
To assess the performance of the Morphing-wing control, measurements of aerodynamics and
inertial forces have been carried out using the Brown University wind-tunnel facility, and a
force sensor1 positioned below the robot’s body. The experiments are focused on the motion
of the morphing-wings, i.e., wings extension and contraction by means of elbow’s motion. The
response of the morphing-controller is shown in Figure 7.2. As previously mentioned, two
scenarios are considered for testing the performance of the control architecture: i) non-flapping
with Vair = 0m/s), and ii) morphing+flapping at f = 2.5Hz with Vair = 5m/s. The term Vair
1Nano17 transducer ATI Industrial Automation, 0.318 gram-force of resolution, http://www.ati-ia.com/
156
7.2 Control performance
(a) (b)
(c) (d)
©Brown University©Brown University
©Brown University©Brown University
(e) (f)
Figure 7.2: Stills of morphing-wings’ control within the wind-tunnel. The wingbeat cycle is
composed by two phases: downstroke and upstroke. a) Beginning of the downstroke. The body of
the specimen is lined up in a straight line, elbow joint is∼ 58o, b) end of downstroke, the membrane
is cambered and the wings are still extended, elbow joint is ∼ 5o, c) middle of downstroke, the
wings are extended to increase lift, elbow joint is ∼ 20o, d) upstroke, the wings are folded to
reduce drag, elbow joint is ∼ 45o. Figures a-b illustrate the process to measure aerodynamics
loads using the force sensor located at the center of mass of the robot (below the body). Figures
c-d illustrate the process to measure inertial forces at the center of mass produced by both wings
(no aerodynamics loads caused by the membrane). e-f) show the beginning of downstroke and the
end of the upstroke without the membrane.
denotes the airspeed of the wind-tunnel. Performance is then evaluated in terms of: i) tracking
accuracy, ii) actuation speed, and iii) SMA fatigue issues.
157
7.2 Control performance
0 2 3 6
0.15
0.3
0.45
0.6
0.75
t [s]
I [A]
0 2 4 60
20
40
60
80
t [s]
3 [deg
]
0.1 0.2 0.3 0.40
25
50
75
t [s]
3 [deg
]
0.1 0.2 0.3 0.40
0.2
0.4
0.6
0.8
t [s]
Posit
ion er
ror
(Anti-slack)~10% of max. current
(Anti-overload)(a) (b)
(c) (d)
q3 at Vair = 0m/sq3,ref q3,ref
q3 at Vair = 0m/sq3 at Vair = 5m/s
error, Vair = 0m/serror,Vair = 5m/s
0 4
q3 [
deg]
q3 [
deg]
Isma
[A]
Figure 7.3: (Experimental) morphing-wings’ control response. a) Tracking of the elbow’s joint
trajectory at f = 2.5Hz, Vair = 0m/s i.e., no-wind. b) Close-up to a wingbeat cycle. The two plots
describe the control tracking regarding: i) Vair = 0m/s (same than plot-a), and ii) Vair = 5m/s.
c) Electrical current Isma delivered to the antagonistic SMA actuators, and regulated by the
anti-slack and anti-overload mechanisms. d) Position tracking errors from plot-b.
7.2.1.1 SMA accuracy and speed
Figure 7.3a-b shows the experimental results regarding the motion tracking of morphing-wing
trajectories at a wingbeat cycle of f = 2.5Hz. In order to analyze the accuracy of the controller,
7.3b details the time-evolution of q3 during a wingbeat cycle (t = 0.4s). In this figure, the
bio-inspired reference trajectory profile is denoted as q3,ref , in which the downstroke phase
takes 0.22s (wings extended), and the upstroke 0.18s (wings folded). Two experiments have
been carried out: i) morphing tracking with Vair = 0, and ii) with Vair = 5m/s. After
several trials, we observed that significant errors in position tracking mostly occurred during the
upstroke phase. The corresponding errors are shown in Figure 7.3d. During the upstroke, drag
forces caused high aerodynamics loads that introduced serious disturbances that are difficult to
completely reject.
One might expect that the Resistance-Motion relationship (see Algorithm 3) provides a
feasible indirect measurement of motion under any condition, but at higher speeds, variations in
SMA electrical resistance are presented during the entire wingbeat cycle. This fact is a problem
since the RM algorithm evaluates this resistance change only at the end of each wingbeat cycle,
158
7.2 Control performance
0
0.005
0.01
0.015
0.02
3 [Nm]
0
0.005
0.01
0.015
0
0.005
0.01
0.015
0
0.005
0.01
0 1 2 3 4
0.01
0.015
0.02
t [min]
3 [Nm]
0 4 8 12 160
0.004
0.008
0.012
0.016
t [min]
3 [Nm]
2.5Hz1.75Hz
1.35Hz1.1Hz
1.3Hz 1.23Hz
(a) (b)
(c)
0 5
0.002
0.004
0.006
0.008
0.005
0.01
0.015
0.02
Figure 7.4: (Experimental) Performance of the SMA actuator for longer periods of actuation:
a) Nominal operation at 1.3[Hz], b) Overloaded operation at 2.5[Hz], c) Output torque peaks
extracted from overloaded response in plot-b).
and this means that the slope correction of the RM function only takes place for the next cycle.
As a consequence, cumulative errors are introduced in the estimation of q3. These errors could
be reduced by introducing a prediction stage within Algorithm ??. In terms of actuation speed,
the implemented control architecture allowed the system to operate successfully at f = 2.5Hz.
7.2.1.2 SMA fatigue issues
Most of the previous experiments have been carried out for short periods of time. However,
contracting and extending the wings at 2.5Hz requires peaks of input power ∼ 3W , which
could cause the SMA to fatigue over time. Unfortunately for longer periods of SMA actuation,
fatigue issues are observed, causing the output torque of the SMA actuators to decrease over
time. As consequence, SMA performance in terms of actuation speed also decreases quickly
as a function of time. Figure 7.4a-b show the measured output torque curve produced by the
SMA actuators at both nominal and overloaded operation modes. The optimal performance
(f = 2 − 2.5[Hz]) can be maintained up to 1.5 minutes without compromising the actuation
speed (at minute 1.5, the reduction in output torque is about 4.5%). Figure 7.4c shows the
measured peaks of torque produced by the elbow joint during rotation. Each peak corresponds
to the four critical points highlighted in plot-b. Table 7.1 summarizes the data.
After 5 minutes of continuos overloaded operation, the wingbeat frequency has decreased
from f = 2.5Hz to 1.1Hz (stabilizing around 1Hz). This corresponds to ∼ 56% of performance
decrease in terms of actuation speed. On the other hand, -under nominal mode- (see Figure
159
7.2 Control performance
Table 7.1: Performance data of SMA actuation for longer periods of time.
Time Output torque Actuation speed % Performance-reduction1
Nominal initial 0.007[Nm] 1.3[Hz] –
16[min] 0.0068[Nm] 1.23[Hz] 5%
initial 0.018[Nm] 2.5[Hz] –
1.5[min] 0.015[Nm] 1.75[Hz] 30%
Overloaded 3[min] 0.011[Nm] 1.35[Hz] 46%
5[min] 0.008[Nm] 1.1[Hz] 56%
1 % of reduction in the actuation speed.
7.4a), results have shown that the SMA actuators tend to stabilize around 1 − 1.2Hz. These
results confirm that constant output torque in a Migamotor actuator can be only maintained
under nominal operation, at SMA contraction speeds of 300ms (96). Once overloaded, the
critical fatigue point is presented about 1.5 minutes of continuos operation, however, it has
been observed that once the SMA actuator is completely cooled, it is able to raise the maximum
operating frequency of 2.5Hz during another 1.5 minutes. Table 7.1 details the performance
data. In particular, methods for removing or reducing fatigue phenomenon must be analyzed.
One of these methods could be based on investigations related to high-frequency responses of
SMAs and the possibility of using high-bandwidth control systems as a possible approach of
eliminating limit cycles. As demonstrated by (9), high-bandwidth force control could be a
solution.
7.2.2 Attitude control
This section presents wind-tunnel experimental results regarding the performance of the pro-
posed attitude controller in terms of:
1. evaluating the accuracy of the backstepping+DAF controller for tracking pitch and roll
references under the presence of external disturbances caused by aerodynamic loads at
airspeeds up to 5ms−1,
2. demonstrating the assumption of incrementing net body forces (Fnet) thanks to the wing
modulation driven by the backstepping+DAF controller.
7.2.2.1 Forward flight
During forward flight, the bat-robot flaps the wings symmetrically at the desired wingbeat
frequency f . The bio-inspired angular trajectories of the wings are shown in figure 4.5. Wing
160
7.2 Control performance
Figure 7.5: Forward flight control. Backstepping+DAF attitude tracking at: a)-b) roll and pitch
tracking with Vair = 5ms−1, c)-d) roll and pitch tracking with Vair = 2ms−1.
modulation produces pitching and rolling torques (τφ, τθ) due to wing inertias cause angular
accelerations on the body. Using the inertial model of the robot, it is possible to determine
both roll and pitch motions that are produced by these angular accelerations. The experiment
carried out in figure 7.5 uses the calculation of roll and pitch motions as input references for
the attitude controller (φd, θd). The goal is to assess the accuracy of the backstepping+DAF
method during the tracking of φd and θd when subjected to aerodynamic loads. As shown
in figure 7.5a-b, the backstepping+DAF is able to maintain both φ, θ oscillating around the
defined set-point. The roll set-point is 0o whereas the pitch set-point is 5.5o. This configuration
allows for the generation of positive inertial thrust that would drive the robot forward. For
this scenario the wind-tunnel airspeed has been set to 5ms−1. Figures 7.5c-d follow the same
procedure with the difference that pitch set-point has been set to 20o and the airspeed has been
decreased to 2ms−1. The insets show the attitude tracking errors caused by aerodynamic loads
that depend on the airspeed.
7.2.2.2 Turning flight
In turning flight, the roll angle must be modified to allow the robot to turn right or left. Figure
7.6 shows experimental results on how the robot behaves during this process. Roll references are
161
7.2 Control performance
disturbance
disturbance
disturbance
disturbance
Figure 7.6: Turning flight control. Backstepping+DAF attitude tracking at: a)-b) roll and pitch
tracking with Vair = 5ms−1, c)-d) roll and pitch tracking with Vair = 2ms−1.
defined of the form φd = a+ bsin(2πft), where a is the roll set-point angle, b is the amplitude
of the oscillation, and f the desired frequency. On the other hand, pitch references have been
set to θd = 0o. In figure 7.6a, the roll reference φd = a + 0.25sin(2πft) determines that the
robot must turn from right to left by following the set-point command: a = 7o (0 < t ≤ 4),
and a = −2.5o (4 < t ≤ 7). Disturbances have been induced into the system aimed at testing
the reliability of the controller. These disturbances are caused by small loss of lift forces that
occur during the contraction process of the wing. This issue can be observed in Figs. 7.6a-c at
the switching point when t = 4s. The loss in lift forces accounted for about 4% due to a loss of
tension of the wing membrane. In this work the loss of tension has not been quantified. This
issue could be solved by improving the anisotropic property of the wing membrane material.
Disturbances of this kind are difficult to immediately reject at high airspeeds (5ms−1), however
the backstepping+DAF controller has shown accurate performance in attenuating the amplitude
of these oscillations at nominal airspeeds of 2ms−1 (cf. figure 7.6c-d). Without the DAF, the
oscillations caused by the disturbances would be higher than those observed here.
162
7.3 Aerodynamics experiments
0 5 10 150
0.5
1
1.5
Angle of attack [deg]
CL an
d CD
0 5 10 154
6
8
10
12
Angle of attack [deg]
L/D
0 5 10 15 204.995
5
5.005
5.01
Measurements
V air [m
/s]
0 5 10 1500.20.40.60.8
11.2
Angle of attack [deg]
L and
D Fo
rces [
N]
CL non-morphingCL
morphing
CD morphing
CD non-morphing
(a) (b)
(c) (d)
L
D
Figure 7.7: (Experimental) Aerodynamics measurements. a) Comparison between lift and drag
coefficients (CL, CD) with and without the motion of the morphing-wings (Vair = 5m/s, wingbeat
frequency of f = 2.5Hz). b) Lift-to-drag ratio (L/D) as a function of the angle of attack (Vair =
5m/s, f = 2.5Hz ). c) Wind-tunnel airspeed measurements (Vair). d) Lift (L) and drag (D)
forces corresponding to plot-a (with morphing).
7.3 Aerodynamics experiments
An initial aerodynamic analysis was previously introduced in Figure 4.10. The goal of the
experiment was to quantify lift and drag coefficient values to include them into the inertial
model as aerodynamic loads. In that experiment, the wings were fixed to the robot at mid-
downstroke, whereas the angle of attack was increased up to 25o. Contrary, this section shows
aerodynamic performance of the entire robot, performing both flapping and morphing-wings’
motions within the wind-tunnel facility. The experiments carried out in Figure 7.7 are focused
on analyzing the changes in lift and drag that are induced by the morphing wing.
Figure 7.7a compares the obtained lift and drag coefficients when the wings of the robot
are: i) -static-, i.e., non-morphing motions, and ii) describing the bio-inspired wing trajectories,
i.e., flapping+morphing motions synchronized at a wingbeat frequency of f = 2.5Hz. The
morphing trajectory described by the elbows’ joints was shown Figure 7.3b, whereas the flapping
trajectory is a simple sinusoidal signal with amplitude of ±60o with f = 2.5Hz. In terms of
lift production, note in Figure 7.7a how the lift coefficient (CL morphing) is increased by about
46% compared to the non-morphing profile. We have observed that the major contribution in
163
7.4 Efficient Flight
increasing lift forces was due to the flapping motion, accounting for about 28%. The morphing-
motion’s contribution accounted for about 18% thanks to wing area maximization during the
downstroke. On the other hand, the drag coefficient is reduced by about 40% by means of
folding the robot’s wings during the upstroke. The angle of attack corresponding to these
measurements is 9o.
Another important measurement corresponds to the lift-to-drag ratio L/D (Figure 7.7b )
which determines the most efficient angle of attack for sustained flight. This angle corresponds
to 9o (straight flight). At this angle, the corresponding lift and drag forces are shown in Figure
7.7d, being the lift force: 0.97N , about 26% more than the bat’s weight-force of 0.77N . Table
7.2 details these results.
Table 7.2: Lift and drag measurements for an angle of attack of 10o and Vair = 5m/s
Lift force L[N ] Drag force D[N ] CL CD
non-morphing 0.67 0.17 1.03 0.26
flapping+morphing (f = 2.5Hz) 0.97 0.099 1.5 0.152
7.4 Efficient Flight
Wing inertia information is contained in the DAF with the purpose of including desired roll
and pitch angular accelerations within the control law. DAF allows for the proper modulation
of wing kinematics, which impacts the generation of both inertial and aerodynamic forces.
Therefore, the following experiments are aimed at comparing the attitude response of the system
with and without the DAF function, showing the benefits of the proposed controller in terms
of disturbance rejection and net force production. Figure 7.8 quantifies the improvement in
forward flight by carrying out measurements of: i) attitude tracking (φ, θ), ii) wing modulation
(q3), and iii) net force production (Fnet). Note how the backstepping+DAF is able to reject
disturbances caused by increasing the wind-tunnel airspeed up to 5ms−1 (top plots). Like the
experiments in figure 7.5, accurate roll and pitch tracking is ensured during forward and turning
flight. Each controller (with and without DAF) produce a different pattern of wing modulation
(q3) (middle plots). With DAF the upstroke portion of the wingbeat cycle generates less drag
thanks to the fact that the elbow joint contracts sufficiently to reduce the wing area at minimum
span. This clearly affects the value of Fnet and the inertial thrust components (fxb).Finally,
note in the bottom plots how with DAF the bias of net forces Fnet is about 23% higher thanks
to the proper modulation of the wing kinematics. Also, it is confirmed that inertial thrust
164
7.4 Efficient Flight
1 2 3t (s)
00
t (s)
t (s) t (s)
t (s) t (s)(a) (b)
dowstroke upstrokedowstroke upstroke
dowstroke upstroke
Figure 7.8: (Forward flight) benefits of the DAF to the proper modulation of wing-morphology
aimed at incrementing net forces (f = 2Hz, φref = 0o, θref = 10o): a) without the DAF, b)
with the DAF. Top: attitude tracking error and disturbance rejection; middle: detailed wing
modulation (elbow joint q3); bottom: net forces generated.
Table 7.3: List of parameters used for experiments in figure 7.8.
Backstepping AoA Vair f1 FL FD fzb2 fxb Fnet bias
with DAF 9o 5ms−1 1.5− 2.5Hz ¯0.97N ¯0.099N 0.77N [−4,+9]mN 0.11N
without DAF 9o 5ms−1 1.5− 2.5Hz ¯0.97N ¯0.12N 0.77N [−4.5,+7.5]mN 0.09N
1 applies for that range of wingbeat frequencies.2 Mb = 79g, no-battery included.
is positive during the upstroke and negative during the downstroke, causing the net force to
increase or decrease as a function of the wingstroke motion (see inset in figure 7.8b). Table 7.3
summarises the numerical data.
In the experiments of Figure 7.8, efficient flight was evaluated by comparing the difference
between wing modulation profiles (q3) and their impact into the generation of net forces (Fnet).
As mentioned, the differences were produced by comparing the response of the backstepping
attitude control with and without the incorporation of DAF terms into the control law. However
165
7.4 Efficient Flight
xo[m] yo[m]
Zo[m
]
(a)
(b)
(c)
(d)
(e)
q3 no-DAF-1 q3 no-DAF-2
q4 no-DAF-1 q4 no-DAF-2
0.022
0.0215
0.021
0.0205
0.02
Figure 7.9: Effects of different wing modulation profiles (no-DAF terms presented) on lift and
drag production (f = 2.5Hz, Vair = 5ms−1): a) Cartesian trajectory of the wingtip generated
during a wingbeat cycle measured with respect to the base frame {0}, b) wing modulation profile of
elbow joint q3 for different backstepping parameter values of λ2 and λ4 (no-DAF-1: λ2 = λ4 = 0.1,
no-DAF-2: λ2 = λ4 = 0.05), c) wing modulation profile of wrist joint q4 corresponding to the
same backstepping parameter values configuration from plot-b, d) (left) net forces and (right)
lift and drag coefficients generated with the backstepping parameter configuration no-DAF-1, e)
left) net forces and (right) lift and drag coefficients generated with the backstepping parameter
configuration no-DAF-2.
in both cases, the backstepping parameters shown in Table 6.2 were used (λ1, C1, λ2). The goal
of the following set of experiments in Figure 7.9 consists on observing the effects of backstepping
parameter tuning for the generation of wing modulation commands and also show how the
performance of flight is drastically decreased when DAF terms are not included and also the
166
7.4 Efficient Flight
Table 7.4: Summary of performance of backstepping+DAF control and its influence into wing
modulation.
Backstepping AoA Vair f1 FL FD fzb2 Fnet bias λ2, λ4
with DAF3 9o 5ms−1 1.5− 2.5Hz ¯0.97N ¯0.099N 0.77N 0.11N 0.02
without DAF4 9o 5ms−1 1.5− 2.5Hz ¯0.97N ¯0.12N 0.77N 0.09N 0.02
no-DAF-25 9o 5ms−1 1.5− 2.5Hz ¯0.71N ¯0.17N 0.77N 0.085N 0.05
no-DAF-16 9o 5ms−1 1.5− 2.5Hz ¯0.31N ¯0.12N 0.77N 0.021N 0.1
1 applies for that range of wingbeat frequencies.2 Mb = 79g, no-battery included.3,4 data from Figures 7.8 and 7.7.5,6 data from Figure 7.9.
original backstepping parameters are not suitable tuned.
In relation to the backstepping parameter tuning, it has turned out that the parameter
λ2 (roll control) and λ4 (pitch control); cf. Eq. (6.29) is key for the proper tuning of back-
stepping control response. Both parameters directly affect the error e2, having an impact into
the regulation of the joint speed. Whether λ2 and λ4 increase, the slope of the joint angular
function also increases, which consequently enable faster downstroke cycles. This can be ob-
served in the experiments from Figure 7.8 (medium plots) note both q3 profiles tend to have
similar downstroke and upstrokes periods and even with the DAF terms included (Figure 7.8b-
medium) downstroke phases take longer than the upstroke. This response is induced aimed at
incrementing lift forces when wings are extended during the downstroke and reduce drag when
wing are folded during the upstroke. In the practise, backstepping+DAF parameters has been
tuned as λ2 = λ4 = 0.02 aimed at inducing the aforementioned response. Contrary, in Figures
7.9b-c the parameters λ2 and λ4 have been increased: no-DAF-2 refers to λ2 = λ4 = 0.05 and
no-DAF-1 refers to λ2 = λ4 = 0.1. One can note that larger values for λ2 and λ4 induce faster
downstroke phases. Because wing modulation drastically affects both inertial and aerodynamic
responses, note in Figures 7.9d-e how net forces (Fnet) and lift coefficient (CL) profiles dras-
tically decreased compared to the experiments in Figure 7.8a-b (lower plots). The decrease
in performance is even worst because drag coefficients (CD) actually increase a bit, causing
aerodynamic frictions to increase. In conclusion, efficient flight can be achieved when original
backstepping control law is augmented with DAF terms (backstepping+DAF) and both λ2 and
λ4 parameters are lower. The influence of wing-inertia control into lift and net force production
can be really appreciated in Table 7.4 (this table completes the numerical values from 7.3 in
more detail).
167
7.5 Discussion of results: Towards efficient flight
7.5 Discussion of results: Towards efficient flight
7.5.1 Morphing-wing modulation
Taking inspiration from nature, and in particular the morphology and flight kinematics of
bats, we have proposed a biomechanical and control system design that takes advantage of the
morphing-wing capability of the bat flight apparatus. Although a conventional servo system is
used for the primary flapping mechanism, the proposed control enables the bat robot to perform
bio-inspired morphing-wing motions using Shape Memory Alloys used as artificial muscles for
wing retraction and extension. Experiments were carried out to analyze how to properly speed-
up the operation of the SMAs to ensure their feasible use for the application at hand.
In terms of control, the adapted anti-slack and anti-overload mechanisms proved to effec-
tively work in a position control scheme, by servo’ing SMA electrical resistance changes to
accurately estimate the morphing-motion of the wings. Thanks to the implemented Resistance-
Motion (RM) relationship, both mechanisms were analyzed and experimentally adjusted for
regulating the amount of input heating power to be delivered to the SMA artificial muscles.
The fact that our robot does not make use of any motion sensor to provide the angular po-
sition feed-back, and yet achieves satisfactory tracking errors (even in the presence of high
aerodynamics loads), represents a validation of this control approach.
7.5.2 Wing inertia for efficient flight
The results presented in this section demonstrate how the wings can considerably affect the
dynamics/aerodynamics of flight and how to take advantage of wing inertia information to
properly change wings’ geometry during flapping. This fact has been carefully modelled and
quantified for the prototype at hand. It has also been shown how an effective attitude control of
the bat-like robot can be achieved by changing wings’ kinematics in order to generate controlled
inertial forces. The robot’s body mass influence in the generation of pitching and rolling torques
has been quantified, and scaling factors that relate how both inertial quantities increase as
a function of the flapping frequency, were found. These factors allowed the tuning of the
backstepping+DAF control parameters, improving the attitude tracking against high external
disturbances produced by aerodynamic loads.
Moreover, the proposed control strategy was developed and tested for demonstrating the
assumption of incrementing net body forces thanks to the wing modulation driven by the
backstepping+DAF controller. Such increment is significant, about 23%. Part of that increment
168
7.5 Discussion of results: Towards efficient flight
0 1002.5
3
3.5
4
4.5
5
A x,b[m
/s2 ]
Percentage of wingstroke %
0 100Percentage of wingstroke %
A xbms
−2⎡ ⎣
⎤ ⎦
2.5
3
3.5
4
4.5
5
Figure 7.10: (Experimental) comparison of horizontal inertial acceleration (Axb) produced by
the wing modulation at f = 2.5Hz and Vair = 5ms−1: (above) biological data of C. brachyotis
specimen; several measurements reported in (24), (below) BaTboT; several estimations of Axb
based on measurements of inertial thrust fxb.
was provided by the increasing in lift and decreasing of drag forces, cf. Figure 7.7, however,
inertial acceleration or thrust also played and important role in effective net force production.
Figure 7.10 shows the results. For the specimen, cf. Figure 7.10-above, when comparing the
inertial acceleration throughout a wingbeat cycle, calculated as the difference between the
accelerations of the body markers and the centre of mass of the specimen, bats showed large
differences in horizontal peak inertial accelerations between low and high speed flights.
The results support the idea that inertial accelerations produced by the flapping motion of
relatively massive wings can considerably affect the kinematics of bat flight. Horizontal inertial
effects were maximal in both peak and mean accelerations at slow speeds for the downstroke as
well as for the upstroke phases of the wingbeat cycle and decreased as flight speed increased.
One possible explanation for the decrease in horizontal inertial acceleration with speed is that
the total horizontal excursion of the wing decreases with speed as a consequence of a more
vertical stroke plane angle observed at higher speeds. Thus, inertial effects can have important
implications on the way we interpret horizontal accelerations, particularly during slow flights,
cf. Figure 7.10-below.
169
8
Conclusions and Future Work
”Bat flight is fascinating, the engineering journey to accomplish it with BaTboT, even more”
8.1 General conclusions
BaTboT is the first prototype of its kind with the potential for achieving autonomous and sus-
tained flight. All the novel methodologies introduced in this thesis are aimed at achieving that
goal. Motivated by the potential behind bat flight and the lack of highly articulated morphing-
wing MAVs (not necessarily bat-like), BaTboT is definitively a step towards a new generation of
Micro Aerial Vehicles with tremendous dexterity and manoeuvrability that changing the wing’s
geometry enables. In pursuing this long-term vision, a hypothesis was declared:
Quantifying the effects of wing inertia in terms of thrust and lift production and
therefore including wing inertia information into the flight controller will allow
for the proper modulation of wing kinematics that finally would produce and
increase of net forces, thereby improving on flight efficiency.
To demonstrate and validate the aforementioned hypothesis, most of the methods for design,
modelling and control were based on an exhaustive and unprecedented analysis of its biological
counterpart, that finally provided the robust foundation to approach each state of BaTboT’s
development.
170
8.1 General conclusions
Design
A novel design framework relating kinematics/aerodynamics parameters with morphological
parameters was defined based on validated biological data, cf. Table 3.6. It shows how body
and wing mass influence on the proper definition of design criteria. Because BaTboT is the
first bat-like robot with highly articulated wings, this framework is key for future developments
of similar bat-like robots with different morphological parameters.
Modeling
Models for kinematics, dynamics, aerodynamics and actuation were defined and experimentally
validated. An inertial model allowed for the quantification of the influence of wing inertia into
robot’s maneuverability and the key role of proper wing modulation aimed at the production
of rolling and pitching torques for turning and forward flight.
Control
A Flight Control Architecture was defined. The proposed backstepping+DAF method for
attitude control has demonstrated to be key for achieving proper response in terms of attitude
stabilisation and tracking. More important, the assumption of incrementing net body forces
thanks to the wing modulation driven by the backstepping+DAF controller was proved. Such
increment was significant, about 23%. Also, aerodynamics were dramatically improved thanks
to the morphing-wing control, cf. Figure 7.7. In terms of lift production, the lift coefficient
was increased by about 46% thanks to the maximisation of the wing area during extension and
drag was reduced by about 40% by means of folding the robot’s wings during the upstroke.
Mimicking the way bats take advantage of inertial and aerodynamical forces produced by the
wings in order to both increase lift and maneuver is a promising way to design more efficient
flapping wings MAVs. This is a key factor for their effective use in practical applications where
the extremely low payload capacity limits the autonomy operation of flying machines in outdoor
scenarios.
The novel wing modulation strategy and attitude control methodology presented and vali-
dated in this thesis provide a totally new way of controlling flying robots that eliminates the need
of appendices such as flaps and rudders. These developments are a key step towards achieving
the first bat-like robot capable of sustained autonomous flight. The possibility of controlling the
171
8.2 Future Work
shape of the wings has great potential to improve the maneuverability of current Micro-Aerial
Vehicles. Also, the overall results highlight the importance of the incorporation of inertial effects
in future analyses of the kinematics of flapping locomotion and how inertial-based control for
attitude regulation is useful when having bio-inspired MAVs with an important wing-to-body
mass ratio.
8.2 Future Work
The need of new light and robust materials for robot fabrication is a top order for future
developments of BaTboT. Actually, ABS plastic has resulted too heavy for the application at
hand, therefore constraining the possibility of achieving sustained flight. Including electronics,
BaTboT has a mass of 125g with a wingspan of 53cm. The specimen, on the other hand, has a
mass of ∼ 80g, a proper tradeoff between mass and wingspan. Attempting to decrease BaTboT
mass will require the need of carbon fibre fabrication and the incorporation of light electronics
and sensors. Specially the LiPo battery that contributes about 36.8% of the overall mass and
only provides 800mAh of power, being inefficient for BaTboT requeriments. The development
of a low-mass high-power system based on super-capacitors, for instance, it is required.
The use of Shape Memory Alloys has been key in achieving light actuated morphing wings
but their power consumption and actuation speed are still a radical limitation. This thesis
explored how to speed-up SMA operation while maintaining the limits of power consumption,
however, future work dedicated to improve on SMA performance is required, specially in terms of
eliminating fatigue phenomenon by means of introducing high bandwidth controllers. Methods
for embedding force feedback into a single SMA actuator is a top order for future development
of these smart actuators.
To control wing modulation, further research should be directed to quantify the effects of the
incoming airflow on the wings, with the aim of adjusting wing morphology in a more efficient
way in order to dramatically reduce drag. Actually, the wings of biological bats have tiny hairs
that sense airflow conditions, and there is some evidence that this sensing apparatus in bats
contributes to their flight efficiency (8). Besides elbow’s contraction, bats’ 3-DoF wrist joint
also contributes in folding the digits towards the body. As a consequence, bats can reduce
their wingspan about 70% during the upstroke (28). In this work, we have attempted to mimic
part of that complexity, however, our robot is able to reduce its wingspan about 23% during
172
8.3 Thesis schedule
the upstroke. This mechanical limitation is due to the fact that the wrists’ joints are under-
actuated, contributing about 5% during wings’ contraction. Significant drag reduction still
remains a challenge. Also, further investigations regarding the highly-anisotropic property of
the wing membrane should be taken into account. Furthermore, future wing designs should
incorporate additional sensors, such as a flex sensor bound to the membrane or skeleton. This
will give real-time knowledge of the shape of the wing, allowing the wings’ shape in response
to external variables to be tailored.
8.3 Thesis schedule
The Gantt diagram details the tasks and milestones carried out during the thesis development.
2w1.1) Biomechanics: insects, birds, bats
6w1.2) Bat physiology
4w1.3) Bat aerodynamics
4w1.4) Smart materials
16w1) Nature's Flyers@UPM
3w2.1) Bat morphology
2w2.2) Bat muscles
2w2.3) Bat membranes
3w 1d2.4) Bat aerodynamics
3w 1d2.5) Bat skeleton structure
6w 1d2.6) Initial BAT-concept proposal
19w 3d2) Bat Kinematics of Flight@UPM
3) Milestone 1: Chapter 1-2-3 DocDocument Overview of tasks 1 and 2
2w 4d4.1) Choice of species: P. bachiotelis.
3w 2d 4h4.2) Morphological parameters
4w4.3) Wing kinematics of flight
2w4.4) Dynamics and aerodynamics
5w4.5) Proposed model for wing-muscle actuation
17w 1d 4h4) In-vivo Bat flight analysisIn-situ @Brown
5) Milestone 2: Chapter 4 DocDocument Overview of tasks 4
3w6.1) General morphology and kinematics
3w6.2) Dynamics Modeling
3w6.3) Aerodynamics Modeling
5w 1d6.4) SMA actuation modeling
3w 2d6.5) Open-loop testing
3w6.6) Model validation against in-vivo behavior
20w 3d6) BaTboT ModelingIn-situ @Brown
7) Milestone 3: Chapter 5 DocDocument Overview of tasks 6
8w 1d8.1) CAD development
5w 3d8.2) biomechanics insights
6w 1d8.3) ABS skeleton: body+wings
6w 4d8.4) The wing-membrane6w 4d8.5) Actuators and sensors
6w 4d8.6) Hardware onboard
40w 2d8) BaTboT design and biomechanics@UPM
9) Milestone 4: Chapter 6 DocDocument Overview of tasks 8
19w 2d10.1) Morphing wing control
13w 4d10.2) Flapping wing control
23w 4d10.3) Control architecture
23w 1d10.4) closed-loop testing.
80w 1d10) BaTboT Control@UPM
11) Milestone 5: Chapter 7 DocDocument Overview of tasks 8
38w 1d12.1) Performance analysis (wind-tunnel)
34w12.2) BaTboT model identification
20w 3d12.3) Investigation of 6DoF flight
29w12.4) Flight control and maneuvering.
121w 4d12) Experimental resultsIn-situ @Brown
13) Milestone 6: Final DocDocument Overview of the Project results
Title EffortNature's Flyers @UPM
Biomechanics: insects, birds, bats 100%
Bat physiology 100%
Bat aerodynamics 100%
Smart materials 100%
Bat Kinematics of Flight @UPM
Bat morphology 100%
Bat muscles 100%
Bat membranes 100%
Bat aerodynamics 100%
Bat skeleton structure 100%
Initial BAT-concept proposal 100%
Milestone 1: Chapter 1-2-3 Doc Document Overview of tasks 1 and 2
In-vivo Bat flight analysis In-situ @Brown
Choice of species: P. bachiotelis. 100%
Morphological parameters 100%
Wing kinematics of flight 100%
Dynamics and aerodynamics 100%
Proposed model for wing-muscle actuation 100%
Milestone 2: Chapter 4 Doc Document Overview of tasks 4
BaTboT Modeling In-situ @Brown
General morphology and kinematics 100%
Dynamics Modeling 100%
Aerodynamics Modeling 100%
SMA actuation modeling 100%
Open-loop testing 100%
Model validation against in-vivo behavior 100%
Milestone 3: Chapter 5 Doc Document Overview of tasks 6
BaTboT design and biomechanics @UPM
CAD development 100%
biomechanics insights 100%
ABS skeleton: body+wings 100%
The wing-membrane 100%
Actuators and sensors 100%
Hardware onboard 100%
Milestone 4: Chapter 6 Doc Document Overview of tasks 8
BaTboT Control @UPM
Morphing wing control 100%
Flapping wing control 100%
Control architecture 100%
closed-loop testing. 100%
Milestone 5: Chapter 7 Doc Document Overview of tasks 8
Experimental results In-situ @Brown
Performance analysis (wind-tunnel) 100%
BaTboT model identification 100%
Investigation of 6DoF flight 100%
Flight control and maneuvering. 100%
Milestone 6: Final Doc Document Overview of the Project results
Qtr 4 2009 Qtr 1 2010 Qtr 2 2010 Qtr 3 2010 Qtr 4 2010 Qtr 1 2011 Qtr 2 2011 Qtr 3 2011 Qtr 4 2011 Qtr 1 2012 Qtr 2 2012 Qtr 3 201
173
9
Publications
The development of this thesis has allowed the following scientific production, including JCR
referred journals, book chapters, conference proceedings and press articles.
9.1 Journals, book chapters and conference proceedings
Referred Journals (ISI-JCR)
1. (Q1) Colorado J., Barrientos A., Rossi C., Breuer, K, 2012. Biomechanics of smart wings in a bat robot:
morphing-wings using SMA actuators. Bioinspiration and Biomimetics. vol. 7, 036006 (16pp). (Topic
on BaTboT’s morphing-wing mechanism driven by SMA muscles)
2. (Q4) Colorado J., Barrientos A., Rossi C., 2011. (Bio-inspired Robots with Smart Muscles: Modeling,
Control, and Actuation) Msculos Inteligentes en Robots Biolgicamente Inspirados: Modelado, Control y
Actuacin. Revista Iberoamericana de Automtica e Informtica Industrial, vol. 8, No. 4, 385-396, published
by Elsevier. (Topic on SMA phenomenological modelling and force control)
3. (Q1) Colorado J., Barrientos A., Rossi C., 2012. Inertial Attitude Control of a Bat-like Morphing-wing
Micro Air Vehicle. Bioinspiration and Biomimetics. Accepted. (Topic on flight control using the
backstepping+DAF attitude approach)
4. (Q1) Colorado J., Barrientos A., Rossi C., 2012. Learning from bats: the influence of wing modulation
for efficient robotic flying vehicles. Journal of Royal Society Interface. Accepted. (Topic on kinematics
of bat flight applied to efficient flight)
5. (Q1) Rossi C, Colorado J., Coral W., Barrientos A., 2011. Bending Continuous Structures with SMAs:
a Novel Robotic Fish Design. Bioinspiration and Biomimetics, vol. 6, No. 4, 15pp. (Topic on SMA
identification and resistance control)
6. (Q1) Barrientos A., Colorado J., del-Cerro J., Martinez A., Rossi C., Sanz D., Valente J. 2011. Aerial
Remote Sensing in Agriculture: A practical approach to area coverage and path planning for fleets of mini
aerial robot. Journal of Field Robotics, vol 28, issue 5, pp 667-689. (Topic on backstepping+DAF
attitude control applied to conventional UAV)
Book chapters
1. Coral W., Rossi C., Colorado J., Barrientos, A., 2012. Muscle-like actuation in biologically inspired
robots: The state of arts review on SMA technology, in book Smart actuation and sensing systems -
174
9.2 Press and media
Recent advances and future challenges. In-Tech, ISBN: 978-953-307-990-4. (Literature review on
SMA technology used as actuators in bio-inspired robots
Conferences
1. Colorado J, Rossi C, Coral W, Barrientos A. 2012. SMA-driven modulation of highly articulated wings:
insights on efficient flight inspired by bats. IEEE International Conference on Intelligent Robots and
Systems -IROS2012. Workshop on Smart materials and alternative technologies for bio-inspired robots
and systems, Algarve, Portugal, Oct.7-12.
2. Colorado J., Barrientos A., Rossi C., 2011. Biomechanics of morphing wings in a Bat-robot actuated by
SMA muscles. Proceedings of the International Workshop on bio-inspired robots, Nantes, France, April
6-8.
3. Colorado J. Aerial, Terrestrial and Aquatic Robotics: Navigation and Control Issues. Invited talk at
Pontificia Universidad Javeriana, Cali, Colombia. August 2010.
9.2 Press and media
1. Physics Society USA: http://phys.org/news/2012-06-batbot-imitating-smart-maneuvers.html
2. The Verge USA: http://www.theverge.com/2012/6/4/3060768/batbot-flying-robot-drone
3. ABC.es: http://www.abc.es/20120609/ciencia/abci-crean-robot-vuela-como-201206091312.html
4. RTVE: http://www.rtve.es/noticias/20120130/robot-murcielago-musculos-artificiales/493938.shtml
175
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178
Declaration
I herewith declare that I have produced this thesis document without the prohibited assistance of
third parties and without making use of aids other than those specified; notions taken over directly
or indirectly from other sources have been identified as such.
The thesis work was conducted from 2010 to 2012 under the supervision of Prof. Antonio Barrientos
Cruz, PhD, and Prof. Claudio Rossi, PhD.
Madrid, Spain 2012
10
Annexes
10.1 Floating-base forward kinematics Matlab-code
1 function [Xb,Vb,Vdb] = fb Fkine(DH,q,qd,qdd)
2 %Add a virtual 6DOF kinematic joint, q, qd, add, contain the overall joint trajectory profile of the wing
3 fb q = [0 0 0 0 0 0 q]'; fb qd = [0 0 0 0 0 0 qd]'; fb qdd= [0 0 0 0 0 0 qdd]';
4 %obtain 3x3 rotation matrix that relates joint {1} with the floating base {0}5 c4 = cos(q(4)); s4 = sin(q(4)); c5 = cos(q(5)); s5 = sin(q(5)); c6 = cos(q(6)); s6 = sin(q(6));
6 r = [ c5*c6, c4*s6+s4*s5*c6, s4*s6−c4*s5*c6;7 −c5*s6, c4*c6−s4*s5*s6, s4*c6+c4*s5*s6;
8 s5, −s4*c5, c4*c5 ];
9 p = fb q(1:3);
10 p skew = [ 0, −p(3), p(2);
11 p(3), 0, −p(1);12 −p(2), p(1), 0 ];
13 %coordinate transform from fixed to floating base coordinates
14 Xb = [ r, zeros(3); −r*p skew, r ];
15 % Derivative of r
16 rd = [ 1 0 sin(fb q(5));
17 0 cos(fb q(4)) −sin(fb q(4))*cos(fb q(5));
18 0 sin(fb q(4)) cos(fb q(4))*cos(fb q(5)) ];
19 omega = rd*fb qd(4:6); pd = fb qd(1:3);
20 %spatial velocity of the floating base expressed in fixed−base coordinates.
21 Vb = [ omega; pd+cross(p,omega) ];
22 c4d = −sin(fb q(4))*fb qd(4); s4d = cos(fb q(4))*fb qd(4);
23 c5d = −sin(fb q(5))*fb qd(5); s5d = cos(fb q(5))*fb qd(5);
24 rdd = [ 0 0 s5d;
25 0 c4d −s4d*c5−s4*c5d;26 0 s4d c4d*c5+c4*c5d ];
27 omegad = rd*fb qdd(4:6) + rdd*fb qd (4:6); pdd = fb qdd(1:3);
28 %spatial acceleration of the floating base expressed in fixed−base coordinates.
29 Vdb = [ omegad; pdd+cross(pd,omega)+cross(p,omegad) ];
180
10.2 Floating-base inverse dynamics Matlab-code
10.2 Floating-base inverse dynamics Matlab-code
1 function [FT,Vdb] = IDf(Xb,Vb,Vdb,q,qd,qdd,Faero,flag)
2 % Evaluating for each wing
3 for k=1:2
4 %Floating base contribution
5 Vel(k).V(1)=Vbf;
6 Acel(k).A(1)=Abf+[0 0 0 0 0 −9.81]';7 H=[0;0;1;0;0;0];
8 %Forward recurrence: Spatial Velocity and Acceleration
9 for i=1:n
10 V(i)=Vel(k).V(i);
11 A(i)=Acel(k).A(i);
12 %Calculating 3x3 transformations
13 [r,p]=fkine2(q(i)); %computes r and p with basic forVrd kinematics using Hd−parameters14 %6x6 notation
15 psk=[0 −p(3),p(2);p(3),0,−p(1);−p(2),p(1),0];16 R=[r,zeros(3,3);zeros(3,3),r]; P=[eye(3),−psk;zeros(3,3),eye(3)];17 %Compute Spatial Velocity
18 Vel(k).V(:,i+1)=P*R'*V(:,i)+H*Qd(i);
19 %Spatial Acceleration terms
20 V skew1=[0 −V(3,i+1),V(2,i+1);V(3,i+1),0,−V(1,i+1);−V(2,i+1),V(1,i+1),0]; %actual
21 V skew1 6=[V skew1,zeros(3,3);zeros(3,3),V skew1];
22 Hd=V skew1 6*H;
23 V skew2=[0 −V(3,i),V(2,i);V(3,i),0,−V(1,i);−V(2,i),V(1,i),0];24 V skew2 6=[V skew2,zeros(3,3);zeros(3,3),V skew2];
25 Pd=V skew2 6*P';
26 %Compute Spatial Acceleration
27 Acel(k).A(:,i+1)=P*R'*A(:,i)+Pd*V(:,i)+Hd*qd(i)+H*qdd(i);
28 end
29 %Backward recurrence: spatial forces
30 if flag ==1
31 Force(k).F(n+1)=Faero; %include aerodynamics forces
32 else
33 Force(k).F(n+1)=zeros(6); %no−aero load (just accounting inertial effects
34 end
35 for i=n:−1:136 F(i)=Force(k).F(i);
37 [Jcm,m,s]=inertias(i); %computes the body tensor, mass, and distance to CM
38 s skew=[0 −s(3),s(2);s(3),0,−s(1);−s(2),s(1),0];39 %6x6 notation
40 S=[eye(3) s skew;zeros(3,3) eye(3)];
41 Sd=[zeros(3,3),V skew1 6*s skew;zeros(3,3),zeros(3,3)];
42 I=S'*[Jcm zeros(3,3);zeros(3,3) m(i)*eye(3)]*S;
43 id=V skew1 6*I;
44 %Compute Spatial forces
45 Force(k).F(:,i)=R*P'*F(:,i+1)+I*A(:,i+1)+(Id+I*Sd')*V(:,i+1);
46 end
47 end
48 R0 b =FB trans(q,Xb); %Computes transformation from frame {0} to body frame {b}49 FT=R0 b*Force(1).F(:)+R0 b*Force(2).F(:) %Floating base force FT
50 Ifb=I base() + Xb'*I*Xb; %Floating body inertia
51 Vdb=inv(Ifb)*FT; %Floating body acceleration
10.3 Floating-base forward dynamics Matlab-code
1 function [Abf] = FDf(FT,VdT,q,qd)
181
10.4 SMA phenomenological model Matlab-code
2 % Evaluating for each wing
3 for k=1:2
4 %Compute velocity dependent force terms C(q,qd)qd (using inverse dyn for this purpose)
5 [C tau] = IDf(zeros(6),zeros(6),zeros(6),q(k),qd(k),zeros(n),zeros(6));
6 % Step 1. Compute F = FT(q) − C(q,qd)qd
7 F(k).Ft = FT(k) − C tau;
8 %wing−recurrence9 for i=1:n
10 %6D transformations (R,P)
11 [r,p]=fkine2(q(i)); %computes r and p with basic forVrd kinematics using Hd−parameters12 %6x6 notation
13 psk=[0 −p(3),p(2);p(3),0,−p(1);−p(2),p(1),0];14 R=[r,zeros(3,3);zeros(3,3),r]; P=[eye(3),−psk;zeros(3,3),eye(3)];15 P = R*P';
16 %6D Inertia (I)
17 [Jcm,m,s]=inertias(i); %computes the body tensor, mass, and distance to CM
18 s skew=[0 −s(3),s(2);s(3),0,−s(1);−s(2),s(1),0];19 S=[eye(3) s skew;zeros(3,3) eye(3)];
20 Sd=[zeros(3,3),V skew1 6*s skew;zeros(3,3),zeros(3,3)];
21 I=S'*[Jcm zeros(3,3);zeros(3,3) m(i)*eye(3)]*S;
22 %Filling the high level spatial terms
23 if i<n
24 for a=0:5
25 for b=0:5
26 P h(((6*z)−5)+a,((6*g)+1)+b)=P(a+1,b+1); %P (6nx6x)
27 I h(c1+a,((6*c2)+1)+b)=I(a+1,b+1); %diag{I1,...In}(6nx6n)28 H h(u+a,i)=[0 0 1 0 0 0]'; %diag{H1,...Hn}(6nxn)29 end
30 end
31 z=z−1; g=g−1; u=u+6; c1=c1+6; c2=c2+1;
32 end
33 end
34 % Step 2. Compute the system overall mass M
35 M(k).MT = H h'*inv(P')*I h*inv(P)*H h;
36 % Step 3. Compute joints accelerations of the wings Qdd
37 Qdd(k).QT = inv(M(k).MT)*F(k).Ft;
38 end
39 % Step 4. Projecting the joint acceleration of both wings onto the bat's body
40 R2 1 =FB trans(q,Xbf); %Computes transformation from wing to body frame
41 H base = zeros(6*n,1);
42 H base(1:6) = ones(6,1);
43 Qdd T = H base*(Qdd(1).QT + R2 1*Qdd(2).QT);
44 % Step 5. Projecting spatial accelerations of both wings onto bat's body
45 Vd T = H base*(VdT(1) + R2 1*VdT(2));
46 % Solving the floating base spatial acceleration
47 Abf = Vd T−Qdd T;
10.4 SMA phenomenological model Matlab-code
1 function [T,strain,stress,theta] = SMA phenomenologicalModel(I,step,Time)
2 To = 20; %ambient temperature [C]
3 m = 0.00014; %SMA mass [Kg]
4 R = 8.5; %SMA initial resistance [Ohms]
5 Lo = 0.085; %link length
6 ro = 0.0025; %Link joint radius
7 % Fixed Parameters
8 Cp = 0.2; %Specific heat of wire
9 Ac = 0.0004712; %SMA wire?s circumferential area per unit length (150um)
10 hc = 150; %Heat convection coefficient
182
10.4 SMA phenomenological model Matlab-code
11 t = 0:step:Time; %Time vector
12 %Initial conditions
13 T(1) = To(1); %Initial Temperature [C]
14 stress(1) = 75; %Initial stress [MPa]
15 strain(1) = 0.04; %[MPa]
16 Text = To(1);
17 %**************************************************************************18 %Evolution during Heating
19 %Temperature [C]
20 p = length(t); cont2 = 1; cont = 1;
21 tempo =1;
22 for i=1:p−123 T(i+1) = step*((I*I*R)−hc*Ac*(T(i)−Text))+T(i); %Heating Temperature
24 cont = cont+1;
25 if tempo ≤ (length(To)−1)26 if cont > p/(length(To)−1);27 cont2 =cont2+1;
28 Text = To(cont2);
29 cont = 1;
30 end
31 tempo = tempo+1;
32 end
33 end
34 %Stress computing as a function of temperature
35 As = T(1); Af = T(i+1); aA = pi/(Af−As);36 bA = −aA/10.3; %10.3 is the effect stress constant on Austenite temperatures [MPa.1/C]
37 p = length(T);
38 for j=1:p−139 stress(j+1) = step*(((0.55+1120*(1/(Af−As)))*((T(j+1)−T(j))/step))/(1+1120*(1/(Af−As))))+stress(j);
%Computing stress [MPa]
40 end
41 %Martensite fraction computing and its derivative:
42 p = length(stress);
43 for k=1:p−144 M(k) = 0.5*(cos(aA*(T(k)−As)+bA*stress(k))+1); %Martensite fraction during heating
45 dM(k) = −0.5*(sin(aA*(T(k)−As)+bA*stress(k)))*(aA*((T(k+1)−T(k))/step)+bA*((stress(k+1)−stress(k))/step));46 T h(k) = T(k);
47 end
48 Austenite = M;
49 %Strain computing as a function of stress, temperature, and Marsenite fraction
50 p = length(M);
51 for u=1:p−152 strain(u+1) = (step/75000)*(((stress(u+1)−stress(u))/step)−0.55*((T(u+1)−T(u))/step)+1120*((M(u+1)−M(u))/step))+strain
%Computing strain [MPa]
53 end
54 %Kinematics model (SMA attached to a link)
55 p = length(strain); Theta(1) = 0; ΔY = Lo;
56 for w=1:p−157 Theta(w+1) = (−step*((Lo*((strain(w+1)−strain(w))/step))/(2*ro)))+Theta(w);58 end
59 Theta = Theta*(180/pi);
60 %**************************************************************************61 %Evolution during Cooling
62 %Temperature
63 i = i+1; i flag = i+1;
64 cont = 1; %flag counter used for knowing how many steps are required in cooling phase
65 Ms = T(i);
66 while (T(i) > (To+0.5))
67 T(i+1) = step*(−hc*Ac*(T(i)−To))+T(i); %Cooling temperature
68 t(i+1) = t(i)+step; %Filling time vector with the cooling phase
69 i = i+1; cont = cont+1;
183
10.5 Control code programmed into the Arduino
70 end
71 %Stress computing as a function of temperature
72 Mf = T(i); aM = pi/(Ms−Mf);73 bM = −aM/10.3; %10.3 is the effect stress constant on Austenite temperatures [MPa.1/C]
74 j = j+1;
75 j flag = j+1;
76 for j2=i flag:cont−177 stress(j+1) = step*(((0.55+1120*(1/(Ms−Mf)))*((T(j2+1)−T(j2))/step))/(1+1120*(1/(Ms−Mf))))+stress(j);
%Computing stress [MPa]
78 j = j+1;
79 end
80 %Martensite fraction computing and its derivative:
81 k = k+1; j2 = i flag; k flag = k; temp = 1;
82 for k2=j flag:cont−183 M(k) = 0.5*(cos(aM*(T(j2)−Mf)+bM*stress(k2)))+0.5; %Martensite fraction during heating
84 Martensite(temp) = M(k);
85 T c(temp) = T(j2);
86 j2 = j2+1; k = k+1; temp = temp+1;
87 end
88 %Strain computing as a function of stress, temperature, and Marsenite fraction
89 u = u+1; j2 = i flag; k2 = j flag;
90 for u2=k flag:(cont−3)91 strain(u+1) = (step/28000)*(((stress(k2+1)−stress(k2))/step)−0.55*((T(j2+1)−T(j2))/step)+1120*((M(u2+1)−M(u2))/step))+
%Computing strain [MPa]
92 j2 = j2+1; k2 = k2+1; u = u+1;
93 end
94 end
10.5 Control code programmed into the Arduino
1 //
2 // control.cpp
3 //
4 //
5 // Created by Julian Colorado on 06/04/11.
6 // Copyright 2011 MyCompanyName . All rights reserved.
7 //
8
9 //#include <iostream>
10 #include <Wire.h>
11 #include <Servo.h>
12
13 int gyroResult[3], accelResult[3]; //Store raw sensor output
14 int loopsPerServoUpdate = 2; //Update servos once every loopsPerServoUpdate loops
15 int loopCounter = 0; //Used for driving servos once every loopsPerServoUpdate loops
16 float timeStep = 0.01; //Main loop should run at 1/timeStep Hz
17 float biasGyroX, biasGyroY, biasGyroZ; //Bias values
18 float biasAccelX, biasAccelY, biasAccelZ; //for bias corrections
19 float gyroX, gyroY, gyroZ; //Values after bias
20 float accelX, accelY, accelZ; //correction in degrees
21 float pitchGyro = 0; //Pitch according to gyro
22 float pitchAccel = 0; //Pitch according to accelerometers
23 float pitchPrediction = 0; //Output of Kalman filter
24 float rollGyro = 0; //Roll according to gyro
25 float rollAccel = 0; //Roll according to accelerometers
26 float rollPrediction = 0; //Output of Kalman filter
27 float giroVar = 0.1; //Was 0.1
28 float ΔGiroVar = 0.1; //Was 0.1
29 float accelVar = 5; //Was 5
184
10.5 Control code programmed into the Arduino
30 float Pxx = 0.1; //Angle variance; was 0.1
31 float Pvv = 0.1; //Angle change rate variance; was 0.1
32 float Pxv = 0.1; //Angle and angle change rate covariance; was 0.1
33 float kx, kv; //Mysterious Kalman thingies
34 const int FIRArrayLength = 36; //Number of taps for low pass FIR filter
35 float accelXFIRArray[FIRArrayLength]; //Array for FIR calculations
36 float accelYFIRArray[FIRArrayLength]; //Array for FIR calculations
37 float accelZFIRArray[FIRArrayLength]; //Array for FIR calculations
38 float gyroXFIRArray[FIRArrayLength]; //Array for FIR calculations
39 float gyroYFIRArray[FIRArrayLength]; //Array for FIR calculations
40 float FIRFilter[FIRArrayLength] = { //FIR filter coefficients for: 0−4Hz: gain 5, ripple 12dB; 10−50Hz: gain 0.00000
41 0.00008103385604967716, 0.00027111799242077064, 0.0006854974917591413, 0.0014576515867607135,
42 0.0027530631363864716, 0.004753901748829375, 0.007635997857285217, 0.011541206789928664,
43 0.016544708776664904, 0.022626620444902264, 0.02964895502860427, 0.03734831257674501,
44 0.045342366754130024, 0.053157699234441635, 0.060270435821884946, 0.06616236694645264,
45 0.0703764023237606, 0.07257493449372505, 0.07257493449372505, 0.0703764023237606,
46 0.06616236694645264, 0.060270435821884946, 0.053157699234441635, 0.045342366754130024,
47 0.03734831257674501, 0.02964895502860427, 0.022626620444902264, 0.016544708776664904,
48 0.011541206789928664, 0.007635997857285217, 0.004753901748829375, 0.0027530631363864716,
49 0.0014576515867607135, 0.0006854974917591413, 0.00027111799242077064, 0.00008103385604967716
50 };51 const int zeros = 10;
52 const int poles = 10;
53 float accelXInput[zeros + 1], accelXOutput[poles + 1];
54 float accelYInput[zeros + 1], accelYOutput[poles + 1];
55 float accelZInput[zeros + 1], accelZOutput[poles + 1];
56 float gyroXInput[zeros + 1], gyroXOutput[poles + 1];
57 float gyroYInput[zeros + 1], gyroYOutput[poles + 1];
58 float minServoAngle = 50; //Limits to the throw
59 float maxServoAngle = 130; //of the servo arms
60 unsigned long timer; //Timer for calibrating loop time to timeStep seconds.
61 Servo leftServo, rightServo, throttleServo, rudderServo;
62 float pFactor = 1.5; //Multiplier between angle errors and servo signals
63 float leftCalc, rightCalc, leftServoSignal, rightServoSignal;
64
65 //Function for writing a byte to an I2C device
66 void writeTo(byte device, byte toAddress, byte val) {67 Wire.beginTransmission(device);
68 Wire.send(toAddress);
69 Wire.send(val);
70 Wire.endTransmission();
71 }72 //Function for reading num bytes from an I2C device
73 void readFrom(byte device, byte fromAddress, int num, byte result[]) {74 Wire.beginTransmission(device);
75 Wire.send(fromAddress);
76 Wire.endTransmission();
77 Wire.requestFrom((int)device, num);
78 int i = 0;
79 while(Wire.available()) {80 result[i] = Wire.receive();
81 i++;
82 }83 }84
85 //Function for reading gyroscopes
86 void getGyroscopeReadings(int gyroResult[]) {87 byte buffer[6];
88 readFrom(0x68,0x1D,6,buffer);
89 gyroResult[0] = (((int)buffer[0]) << 8 ) | buffer[1];
90 gyroResult[1] = (((int)buffer[2]) << 8 ) | buffer[3];
185
10.5 Control code programmed into the Arduino
91 gyroResult[2] = (((int)buffer[4]) << 8 ) | buffer[5];
92 }93
94 //Funcion for reading accelerometers
95 void getAccelerometerReadings(int accelResult[]) {96 byte buffer[6];
97 readFrom(0x53,0x32,6,buffer);
98 accelResult[0] = (((int)buffer[1]) << 8 ) | buffer[0];
99 accelResult[1] = (((int)buffer[3]) << 8 ) | buffer[2];
100 accelResult[2] = (((int)buffer[5]) << 8 ) | buffer[4];
101 }102
103 //FIR filter function
104 float FIR(float value, float array[]) {105 int i;
106 float FIROutput = 0;
107 for(i = FIRArrayLength−1; i > 0; i−−){108 array[i] = array[i−1];109 FIROutput += array[i] * FIRFilter[i];
110 }111 array[0] = value;
112 FIROutput += array[0] * FIRFilter[0];
113 return FIROutput;
114 }115
116 //IIR filter function (Low pass Bessel, 5Hz corner freq., 10th order, 100 Hz sample rate;
117 float IIR(float value, float xv[], float yv[]) {118 xv[0] = xv[1]; xv[1] = xv[2]; xv[2] = xv[3]; xv[3] = xv[4]; xv[4] = xv[5]; xv[5] = xv[6]; xv[6] = xv[7]; xv[7] = xv[8]
119 xv[10] = value / 5.681713320e+05;
120 yv[0] = yv[1]; yv[1] = yv[2]; yv[2] = yv[3]; yv[3] = yv[4]; yv[4] = yv[5]; yv[5] = yv[6]; yv[6] = yv[7]; yv[7] = yv[8]
121 yv[10] = (xv[0] + xv[10]) + 10 * (xv[1] + xv[9]) + 45 * (xv[2] + xv[8])
122 + 120 * (xv[3] + xv[7]) + 210 * (xv[4] + xv[6]) + 252 * xv[5]
123 + ( −0.0084477842 * yv[0]) + ( 0.1250487070 * yv[1])
124 + ( −0.8452516757 * yv[2]) + ( 3.4397583115 * yv[3])
125 + ( −9.3457158590 * yv[4]) + ( 17.7418428080 * yv[5])
126 + (−23.8769055880 * yv[6]) + ( 22.5422166190 * yv[7])
127 + (−14.3254341650 * yv[8]) + ( 5.5510863543 * yv[9]);
128 return yv[10];
129 }130
131 void setup() {132 //Variables used for removing zero bias
133 int totalGyroXValues = 0;
134 int totalGyroYValues = 0;
135 int totalGyroZValues = 0;
136 int totalAccelXValues = 0;
137 int totalAccelYValues = 0;
138 int totalAccelZValues = 0;
139 int i;
140
141 //LED on when calibrating
142 //pinMode(13, OUTPUT);
143 //digitalWrite(13, HIGH);
144
145 //Hook up the servos to digital pins 2 thru 5
146 leftServo.attach(5, 600, 2400);
147 rightServo.attach(4, 600, 2400);
148 throttleServo.attach(3, 600, 2400);
149 rudderServo.attach(2, 600, 2400);
150
151 //Configure pins for bat s throttle
186
10.5 Control code programmed into the Arduino
152 pinMode(A0, OUTPUT);
153 digitalWrite(A0, HIGH);
154 pinMode(A1, INPUT);
155 pinMode(A2, OUTPUT);
156 digitalWrite(A2, LOW);
157
158 //Set the servos
159 leftServo.write(90);
160 rightServo.write(90);
161 throttleServo.write(0);
162 rudderServo.write(90);
163 delay(50);
164
165 //Open I2C and serial communications
166 Wire.begin();
167 Serial.begin(115200);
168
169 //Apply settings to sensors
170 writeTo(0x53,0x31,0x09); //Set accelerometer to 11bit, +/−4g171 writeTo(0x53,0x2D,0x08); //Set accelerometer to measure mode
172 writeTo(0x68,0x16,0x1E); //Set gyro to +/−2000deg/sec, 1kHz internal sampling rate and 5Hz low pass filter
173 writeTo(0x68,0x15,0x0A); //Set gyro to 100Hz external sample rate
174
175 //Collect 50 measurements and calculate bias for sensors
176 delay(100);
177 for (i = 0; i < 50; i += 1) {178 getGyroscopeReadings(gyroResult);
179 getAccelerometerReadings(accelResult);
180 totalGyroXValues += gyroResult[0];
181 totalGyroYValues += gyroResult[1];
182 totalGyroZValues += gyroResult[2];
183 totalAccelXValues += accelResult[0];
184 totalAccelYValues += accelResult[1];
185 totalAccelZValues += accelResult[2];
186 delay(50);
187 }188 biasGyroX = totalGyroXValues / 50;
189 biasGyroY = totalGyroYValues / 50;
190 biasGyroZ = totalGyroZValues / 50;
191 biasAccelX = totalAccelXValues / 50;
192 biasAccelY = totalAccelYValues / 50;
193 biasAccelZ = (totalAccelZValues / 50) − 256;
194
195 //LED off when done calibrating
196 digitalWrite(13, LOW);
197 }198
199 void loop() {200 //Initialise the loop timer
201 timer = millis();
202
203 //Set the throttle to the bat
204 throttleServo.write(map(analogRead(1), 0, 1023, 0, 179));
205
206 //Get the raw sensor data
207 getGyroscopeReadings(gyroResult);
208 getAccelerometerReadings(accelResult);
209
210 //Remove bias from raw sensor data and convert to g and deg/s respectively.
211 accelX = (accelResult[0] − biasAccelX) / 256;
212 accelY = (accelResult[1] − biasAccelY) / 256;
187
10.5 Control code programmed into the Arduino
213 accelZ = (accelResult[2] − biasAccelZ) / 256;
214 gyroX = (gyroResult[0] − biasGyroX) / 14.375;
215 gyroY = (gyroResult[1] − biasGyroY) / 14.375;
216
217 //Try to remove the motor and rotor vibrations from the data with a low−pass FIR/IIR filter
218 accelX = IIR(accelX, accelXInput, accelXOutput);
219 accelY = IIR(accelY, accelYInput, accelYOutput);
220 accelZ = IIR(accelZ, accelZInput, accelZOutput);
221 gyroX = IIR(gyroX, gyroXInput, gyroXOutput);
222 gyroY = IIR(gyroY, gyroYInput, gyroYOutput);
223
224 //Calculate angles from the sensor data
225 pitchAccel = atan2(accelY, accelZ) * 360.0 / (2*PI);
226 pitchGyro = pitchGyro + (gyroX * timeStep);
227 pitchPrediction = pitchPrediction + (gyroX * timeStep);
228 rollAccel = atan2(accelX, accelZ) * 360.0 / (2*PI);
229 rollGyro = rollGyro − gyroY * timeStep;
230 rollPrediction = rollPrediction − (gyroY * timeStep);
231
232 //Do the Kalman filter
233 Pxx += timeStep * (2 * Pxv + timeStep * Pvv);
234 Pxv += timeStep * Pvv;
235 Pxx += timeStep * giroVar;
236 Pvv += timeStep * ΔGiroVar;
237 kx = Pxx * (1 / (Pxx + accelVar));
238 kv = Pxv * (1 / (Pxx + accelVar));
239 pitchPrediction += (pitchAccel − pitchPrediction) * kx;
240 rollPrediction += (rollAccel − rollPrediction) * kx;
241 Pxx *= (1 − kx);
242 Pxv *= (1 − kx);
243 Pvv −= kv * Pxv;
244
245 //Use the Kalman output in a controller, limiting the maximum servo arms throw
246 leftCalc = pitchPrediction − rollPrediction;
247 rightCalc = pitchPrediction + rollPrediction;
248 leftServoSignal = 90.0 − (pFactor * leftCalc);
249 rightServoSignal = 90.0 + (pFactor * rightCalc);
250 if (leftServoSignal > maxServoAngle) {leftServoSignal = maxServoAngle;};251 if (leftServoSignal < minServoAngle) {leftServoSignal = minServoAngle;};252 if (rightServoSignal > maxServoAngle) {rightServoSignal = maxServoAngle;};253 if (rightServoSignal < minServoAngle) {rightServoSignal = minServoAngle;};254
255
256 //Controller
257 float pFactor = 3;
258 float minServoAngle = 50;
259 float maxServoAngle = 130;
260 float pitch, roll, leftCalc, rightCalc, leftCalcFIR, rightCalcFIR, leftServoSignal, rightServoSignal;
261 pitch = pitchPrediction;
262 roll = rollPrediction;
263 leftCalc = pitch − roll;
264 rightCalc = pitch + roll;
265 leftCalcFIR = FIR(leftCalc, leftServoFIRArray);
266 rightCalcFIR = FIR(rightCalc, rightServoFIRArray);
267 leftServoSignal = 90.0 − (pFactor * leftCalcFIR);
268 rightServoSignal = 90.0 + (pFactor * rightCalcFIR);
269 if (leftServoSignal > maxServoAngle) {leftServoSignal = maxServoAngle;};270 if (leftServoSignal < minServoAngle) {leftServoSignal = minServoAngle;};271 if (rightServoSignal > maxServoAngle) {rightServoSignal = maxServoAngle;};272 if (rightServoSignal < minServoAngle) {rightServoSignal = minServoAngle;};273 leftServo.write((int)round(leftServoSignal));
188
10.6 Flight Control Matlab environment
274 rightServo.write((int)round(rightServoSignal));
275
276
277 //Update signal every loopsPerServoUpdate loops, aiming for once every 20ms
278 if (++loopCounter ≥ loopsPerServoUpdate) {279 leftServo.write((int)round(leftServoSignal));
280 rightServo.write((int)round(rightServoSignal));
281 loopCounter = 0;
282 }283
284 //Make the loop last timeStep seconds
285 timer = millis() − timer;
286 timer = (int)(round(timeStep * 1000)) − timer;
287 delay(timer);
288 }
10.6 Flight Control Matlab environment
189
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